fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
IsOfFinOrder.pow_inj_mod {n m : ℕ} : x ^ n = x ^ m ↔ n % orderOf x = m % orderOf x :=
hx.pow_eq_pow_iff_modEq | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.pow_inj_mod | null |
@[to_additive]
pow_eq_pow_iff_modEq : x ^ n = x ^ m ↔ n ≡ m [MOD orderOf x] := by
wlog hmn : m ≤ n generalizing m n
· rw [eq_comm, ModEq.comm, this (le_of_not_ge hmn)]
obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le hmn
rw [← mul_one (x ^ m), pow_add, mul_left_cancel_iff, pow_eq_one_iff_modEq]
exact ⟨fun h => Nat.ModEq.add_left _ h, fun h => Nat.ModEq.add_left_cancel' _ h⟩
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_eq_pow_iff_modEq | null |
injective_pow_iff_not_isOfFinOrder : Injective (fun n : ℕ ↦ x ^ n) ↔ ¬IsOfFinOrder x := by
refine ⟨fun h => not_isOfFinOrder_of_injective_pow h, fun h n m hnm => ?_⟩
rwa [pow_eq_pow_iff_modEq, orderOf_eq_zero_iff.mpr h, modEq_zero_iff] at hnm
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | injective_pow_iff_not_isOfFinOrder | null |
pow_inj_mod {n m : ℕ} : x ^ n = x ^ m ↔ n % orderOf x = m % orderOf x := pow_eq_pow_iff_modEq
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_inj_mod | null |
pow_inj_iff_of_orderOf_eq_zero (h : orderOf x = 0) {n m : ℕ} : x ^ n = x ^ m ↔ n = m := by
rw [pow_eq_pow_iff_modEq, h, modEq_zero_iff]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_inj_iff_of_orderOf_eq_zero | null |
infinite_not_isOfFinOrder {x : G} (h : ¬IsOfFinOrder x) :
{ y : G | ¬IsOfFinOrder y }.Infinite := by
let s := { n | 0 < n }.image fun n : ℕ => x ^ n
have hs : s ⊆ { y : G | ¬IsOfFinOrder y } := by
rintro - ⟨n, hn : 0 < n, rfl⟩ (contra : IsOfFinOrder (x ^ n))
apply h
rw [isOfFinOrder_iff_pow_eq_one] at contra ⊢
obtain ⟨m, hm, hm'⟩ := contra
exact ⟨n * m, mul_pos hn hm, by rwa [pow_mul]⟩
suffices s.Infinite by exact this.mono hs
contrapose! h
have : ¬Injective fun n : ℕ => x ^ n := by
have := Set.not_injOn_infinite_finite_image (Set.Ioi_infinite 0) (Set.not_infinite.mp h)
contrapose! this
exact Set.injOn_of_injective this
rwa [injective_pow_iff_not_isOfFinOrder, Classical.not_not] at this
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | infinite_not_isOfFinOrder | null |
finite_powers : (powers a : Set G).Finite ↔ IsOfFinOrder a := by
refine ⟨fun h ↦ ?_, IsOfFinOrder.finite_powers⟩
obtain ⟨m, n, hmn, ha⟩ := h.exists_lt_map_eq_of_forall_mem (f := fun n : ℕ ↦ a ^ n)
(fun n ↦ by simp [mem_powers_iff])
refine isOfFinOrder_iff_pow_eq_one.2 ⟨n - m, tsub_pos_iff_lt.2 hmn, ?_⟩
rw [← mul_left_cancel_iff (a := a ^ m), ← pow_add, add_tsub_cancel_of_le hmn.le, ha, mul_one]
@[to_additive (attr := simp)] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | finite_powers | null |
infinite_powers : (powers a : Set G).Infinite ↔ ¬ IsOfFinOrder a := finite_powers.not | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | infinite_powers | null |
@[to_additive /-- See also `addOrder_eq_card_multiples`. -/]
Nat.card_submonoidPowers : Nat.card (powers a) = orderOf a := by
classical
by_cases ha : IsOfFinOrder a
· exact (Nat.card_congr (finEquivPowers ha).symm).trans <| by simp
· have := (infinite_powers.2 ha).to_subtype
rw [orderOf_eq_zero ha, Nat.card_eq_zero_of_infinite] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Nat.card_submonoidPowers | See also `orderOf_eq_card_powers`. |
@[to_additive (attr := simp) /-- Inverses of elements of finite additive order
have finite additive order. -/]
isOfFinOrder_inv_iff {x : G} : IsOfFinOrder x⁻¹ ↔ IsOfFinOrder x := by
simp [isOfFinOrder_iff_pow_eq_one]
@[to_additive] alias ⟨IsOfFinOrder.of_inv, IsOfFinOrder.inv⟩ := isOfFinOrder_inv_iff
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinOrder_inv_iff | Inverses of elements of finite order have finite order. |
orderOf_dvd_iff_zpow_eq_one : (orderOf x : ℤ) ∣ i ↔ x ^ i = 1 := by
rcases Int.eq_nat_or_neg i with ⟨i, rfl | rfl⟩
· rw [Int.natCast_dvd_natCast, orderOf_dvd_iff_pow_eq_one, zpow_natCast]
· rw [dvd_neg, Int.natCast_dvd_natCast, zpow_neg, inv_eq_one, zpow_natCast,
orderOf_dvd_iff_pow_eq_one]
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_dvd_iff_zpow_eq_one | null |
orderOf_inv (x : G) : orderOf x⁻¹ = orderOf x := by simp [orderOf_eq_orderOf_iff]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_inv | null |
orderOf_dvd_sub_iff_zpow_eq_zpow {a b : ℤ} : (orderOf x : ℤ) ∣ a - b ↔ x ^ a = x ^ b := by
rw [orderOf_dvd_iff_zpow_eq_one, zpow_sub, mul_inv_eq_one] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_dvd_sub_iff_zpow_eq_zpow | null |
@[to_additive (attr := norm_cast)]
orderOf_coe (a : H) : orderOf (a : G) = orderOf a :=
orderOf_injective H.subtype Subtype.coe_injective _
@[to_additive (attr := simp)] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_coe | null |
orderOf_mk (a : G) (ha) : orderOf (⟨a, ha⟩ : H) = orderOf a := (orderOf_coe _).symm | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_mk | null |
@[to_additive mod_addOrderOf_zsmul]
zpow_mod_orderOf (x : G) (z : ℤ) : x ^ (z % (orderOf x : ℤ)) = x ^ z :=
calc
x ^ (z % (orderOf x : ℤ)) = x ^ (z % orderOf x + orderOf x * (z / orderOf x) : ℤ) := by
simp [zpow_add, zpow_mul, pow_orderOf_eq_one]
_ = x ^ z := by rw [Int.emod_add_mul_ediv]
@[to_additive (attr := simp) zsmul_smul_addOrderOf] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | zpow_mod_orderOf | null |
zpow_pow_orderOf : (x ^ i) ^ orderOf x = 1 := by
by_cases h : IsOfFinOrder x
· rw [← zpow_natCast, ← zpow_mul, mul_comm, zpow_mul, zpow_natCast, pow_orderOf_eq_one, one_zpow]
· rw [orderOf_eq_zero h, _root_.pow_zero]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | zpow_pow_orderOf | null |
IsOfFinOrder.zpow (h : IsOfFinOrder x) {i : ℤ} : IsOfFinOrder (x ^ i) :=
isOfFinOrder_iff_pow_eq_one.mpr ⟨orderOf x, h.orderOf_pos, zpow_pow_orderOf⟩
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.zpow | null |
IsOfFinOrder.of_mem_zpowers (h : IsOfFinOrder x) (h' : y ∈ Subgroup.zpowers x) :
IsOfFinOrder y := by
obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp h'
exact h.zpow
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.of_mem_zpowers | null |
orderOf_dvd_of_mem_zpowers (h : y ∈ Subgroup.zpowers x) : orderOf y ∣ orderOf x := by
obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp h
rw [orderOf_dvd_iff_pow_eq_one]
exact zpow_pow_orderOf | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_dvd_of_mem_zpowers | null |
smul_eq_self_of_mem_zpowers {α : Type*} [MulAction G α] (hx : x ∈ Subgroup.zpowers y)
{a : α} (hs : y • a = a) : x • a = a := by
obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp hx
rw [← MulAction.toPerm_apply, ← MulAction.toPermHom_apply, MonoidHom.map_zpow _ y k,
MulAction.toPermHom_apply]
exact Function.IsFixedPt.perm_zpow (by exact hs) k -- Porting note: help elab'n with `by exact` | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | smul_eq_self_of_mem_zpowers | null |
vadd_eq_self_of_mem_zmultiples {G : Type*} [AddGroup G] {x y : G} {α : Type*}
[AddAction G α] (hx : x ∈ AddSubgroup.zmultiples y) {a : α} (hs : y +ᵥ a = a) : x +ᵥ a = a :=
@smul_eq_self_of_mem_zpowers (Multiplicative G) _ _ _ α _ hx a hs
attribute [to_additive existing] smul_eq_self_of_mem_zpowers
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | vadd_eq_self_of_mem_zmultiples | null |
IsOfFinOrder.mem_powers_iff_mem_zpowers (hx : IsOfFinOrder x) :
y ∈ powers x ↔ y ∈ zpowers x :=
⟨fun ⟨n, hn⟩ ↦ ⟨n, by simp_all⟩, fun ⟨i, hi⟩ ↦ ⟨(i % orderOf x).natAbs, by
dsimp only
rwa [← zpow_natCast, Int.natAbs_of_nonneg <| Int.emod_nonneg _ <|
Int.natCast_ne_zero_iff_pos.2 <| hx.orderOf_pos, zpow_mod_orderOf]⟩⟩
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.mem_powers_iff_mem_zpowers | null |
IsOfFinOrder.powers_eq_zpowers (hx : IsOfFinOrder x) : (powers x : Set G) = zpowers x :=
Set.ext fun _ ↦ hx.mem_powers_iff_mem_zpowers
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.powers_eq_zpowers | null |
IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf [DecidableEq G] (hx : IsOfFinOrder x) :
y ∈ zpowers x ↔ y ∈ (Finset.range (orderOf x)).image (x ^ ·) :=
hx.mem_powers_iff_mem_zpowers.symm.trans hx.mem_powers_iff_mem_range_orderOf | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf | null |
@[to_additive
/-- See `AddSubgroup.closure_toAddSubmonoid_of_finite` for a version for finite additive groups. -/]
Subgroup.closure_toSubmonoid_of_isOfFinOrder {s : Set G} (hs : ∀ x ∈ s, IsOfFinOrder x) :
(closure s).toSubmonoid = Submonoid.closure s := by
refine le_antisymm ?_ (le_closure_toSubmonoid s)
rw [closure_toSubmonoid]
refine Submonoid.closure_le.mpr <| Set.union_subset Submonoid.subset_closure
fun x (hx : x⁻¹ ∈ s) ↦ ?_
apply Submonoid.powers_le.mpr (Submonoid.subset_closure hx)
simp [(hs _ hx).mem_powers_iff_mem_zpowers] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Subgroup.closure_toSubmonoid_of_isOfFinOrder | See `Subgroup.closure_toSubmonoid_of_finite` for a version for finite groups. |
@[to_additive /-- The equivalence between `Fin (addOrderOf a)` and
`Subgroup.zmultiples a`, sending `i` to `i • a`. -/]
noncomputable finEquivZPowers (hx : IsOfFinOrder x) :
Fin (orderOf x) ≃ zpowers x :=
(finEquivPowers hx).trans <| Equiv.setCongr hx.powers_eq_zpowers
@[to_additive] | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | finEquivZPowers | The equivalence between `Fin (orderOf x)` and `Subgroup.zpowers x`, sending `i` to `x ^ i`. |
finEquivZPowers_apply (hx : IsOfFinOrder x) {n : Fin (orderOf x)} :
finEquivZPowers hx n = ⟨x ^ (n : ℕ), n, zpow_natCast x n⟩ := rfl
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | finEquivZPowers_apply | null |
finEquivZPowers_symm_apply (hx : IsOfFinOrder x) (n : ℕ) :
(finEquivZPowers hx).symm ⟨x ^ n, ⟨n, by simp⟩⟩ =
⟨n % orderOf x, Nat.mod_lt _ hx.orderOf_pos⟩ := by
rw [finEquivZPowers, Equiv.symm_trans_apply]; exact finEquivPowers_symm_apply _ n
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | finEquivZPowers_symm_apply | null |
pow_finEquivZPowers_symm_apply (hx : IsOfFinOrder x) (a : Subgroup.zpowers x) :
x ^ ((finEquivZPowers hx).symm a : ℕ) = a := by
simpa only [finEquivZPowers_apply] using
congr_arg Subtype.val ((finEquivZPowers hx).apply_symm_apply a) | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_finEquivZPowers_symm_apply | null |
@[to_additive /-- Elements of finite additive order are closed under addition. -/]
IsOfFinOrder.mul (hx : IsOfFinOrder x) (hy : IsOfFinOrder y) : IsOfFinOrder (x * y) :=
(Commute.all x y).isOfFinOrder_mul hx hy | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.mul | Elements of finite order are closed under multiplication. |
@[to_additive]
isMulTorsionFree_iff_not_isOfFinOrder :
IsMulTorsionFree G ↔ ∀ ⦃a : G⦄, a ≠ 1 → ¬ IsOfFinOrder a where
mp _ _ := not_isOfFinOrder_of_isMulTorsionFree
mpr hG := by
refine ⟨fun n hn a b hab ↦ ?_⟩
rw [← div_eq_one] at hab ⊢
simp only [← div_pow, isOfFinOrder_iff_pow_eq_one] at hab hG
exact of_not_not fun hab' ↦ hG hab' ⟨n, hn.bot_lt, hab⟩
@[to_additive]
alias ⟨_, IsMulTorsionFree.of_not_isOfFinOrder⟩ := isMulTorsionFree_iff_not_isOfFinOrder
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isMulTorsionFree_iff_not_isOfFinOrder | null |
not_isMulTorsionFree_iff_isOfFinOrder :
¬ IsMulTorsionFree G ↔ ∃ a ≠ (1 : G), IsOfFinOrder a := by
simp [isMulTorsionFree_iff_not_isOfFinOrder] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | not_isMulTorsionFree_iff_isOfFinOrder | null |
@[to_additive]
sum_card_orderOf_eq_card_pow_eq_one [Fintype G] [DecidableEq G] (hn : n ≠ 0) :
∑ m ∈ divisors n, #{x : G | orderOf x = m} = #{x : G | x ^ n = 1} := by
refine (Finset.card_biUnion ?_).symm.trans ?_
· simp +contextual [Set.PairwiseDisjoint, Set.Pairwise, disjoint_iff, Finset.ext_iff]
· congr; ext; simp [hn, orderOf_dvd_iff_pow_eq_one]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | sum_card_orderOf_eq_card_pow_eq_one | null |
orderOf_le_card_univ [Fintype G] : orderOf x ≤ Fintype.card G :=
Finset.le_card_of_inj_on_range (x ^ ·) (fun _ _ ↦ Finset.mem_univ _) pow_injOn_Iio_orderOf
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_le_card_univ | null |
orderOf_le_card [Finite G] : orderOf x ≤ Nat.card G := by
obtain ⟨⟩ := nonempty_fintype G
simpa using orderOf_le_card_univ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_le_card | null |
@[to_additive]
isOfFinOrder_of_finite (x : G) : IsOfFinOrder x := by
by_contra h; exact infinite_not_isOfFinOrder h <| Set.toFinite _ | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinOrder_of_finite | null |
@[to_additive /-- This is the same as `IsOfFinAddOrder.addOrderOf_pos` but with one fewer explicit
assumption since this is automatic in case of a finite cancellative additive monoid. -/]
orderOf_pos (x : G) : 0 < orderOf x := (isOfFinOrder_of_finite x).orderOf_pos | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_pos | This is the same as `IsOfFinOrder.orderOf_pos` but with one fewer explicit assumption since this
is automatic in case of a finite cancellative monoid. |
@[to_additive /-- This is the same as `addOrderOf_nsmul'` and
`addOrderOf_nsmul` but with one assumption less which is automatic in the case of a
finite cancellative additive monoid. -/]
orderOf_pow (x : G) : orderOf (x ^ n) = orderOf x / gcd (orderOf x) n :=
(isOfFinOrder_of_finite _).orderOf_pow ..
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_pow | This is the same as `orderOf_pow'` and `orderOf_pow''` but with one assumption less which is
automatic in the case of a finite cancellative monoid. |
mem_powers_iff_mem_range_orderOf [DecidableEq G] :
y ∈ powers x ↔ y ∈ (Finset.range (orderOf x)).image (x ^ ·) :=
Finset.mem_range_iff_mem_finset_range_of_mod_eq' (orderOf_pos x) <| pow_mod_orderOf _ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | mem_powers_iff_mem_range_orderOf | null |
@[to_additive
/-- The equivalence between `Submonoid.multiples` of two elements `a, b` of the same additive
order, mapping `i • a` to `i • b`. -/]
noncomputable powersEquivPowers (h : orderOf x = orderOf y) : powers x ≃ powers y :=
(finEquivPowers <| isOfFinOrder_of_finite _).symm.trans <|
(finCongr h).trans <| finEquivPowers <| isOfFinOrder_of_finite _
@[to_additive (attr := simp)] | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | powersEquivPowers | The equivalence between `Submonoid.powers` of two elements `x, y` of the same order, mapping
`x ^ i` to `y ^ i`. |
powersEquivPowers_apply (h : orderOf x = orderOf y) (n : ℕ) :
powersEquivPowers h ⟨x ^ n, n, rfl⟩ = ⟨y ^ n, n, rfl⟩ := by
rw [powersEquivPowers, Equiv.trans_apply, Equiv.trans_apply, finEquivPowers_symm_apply, ←
Equiv.eq_symm_apply, finEquivPowers_symm_apply]
simp [h] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | powersEquivPowers_apply | null |
@[to_additive]
orderOf_eq_card_powers : orderOf x = Fintype.card (powers x : Submonoid G) :=
(Fintype.card_fin (orderOf x)).symm.trans <|
Fintype.card_eq.2 ⟨finEquivPowers <| isOfFinOrder_of_finite _⟩ | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_card_powers | null |
isOfFinOrder_iff_isUnit [Monoid G] [Finite Gˣ] {x : G} : IsOfFinOrder x ↔ IsUnit x := by
use IsOfFinOrder.isUnit
rintro ⟨u, rfl⟩
rw [Units.isOfFinOrder_val]
apply isOfFinOrder_of_finite
alias ⟨_, IsUnit.isOfFinOrder⟩ := isOfFinOrder_iff_isUnit | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | isOfFinOrder_iff_isUnit | null |
orderOf_eq_zero_iff_eq_zero {G₀ : Type*} [GroupWithZero G₀] [Finite G₀] {a : G₀} :
orderOf a = 0 ↔ a = 0 := by
have : Finite G₀ˣ := .of_injective _ Units.val_injective
simp [isOfFinOrder_iff_isUnit] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq_zero_iff_eq_zero | null |
@[to_additive]
zpow_eq_one_iff_modEq {n : ℤ} : x ^ n = 1 ↔ n ≡ 0 [ZMOD orderOf x] := by
rw [Int.modEq_zero_iff_dvd, orderOf_dvd_iff_zpow_eq_one]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | zpow_eq_one_iff_modEq | null |
zpow_eq_zpow_iff_modEq {m n : ℤ} : x ^ m = x ^ n ↔ m ≡ n [ZMOD orderOf x] := by
rw [← mul_inv_eq_one, ← zpow_sub, zpow_eq_one_iff_modEq, Int.modEq_iff_dvd, Int.modEq_iff_dvd,
zero_sub, neg_sub]
@[to_additive (attr := simp)] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | zpow_eq_zpow_iff_modEq | null |
injective_zpow_iff_not_isOfFinOrder : (Injective fun n : ℤ => x ^ n) ↔ ¬IsOfFinOrder x := by
refine ⟨?_, fun h n m hnm => ?_⟩
· simp_rw [isOfFinOrder_iff_pow_eq_one]
rintro h ⟨n, hn, hx⟩
exact Nat.cast_ne_zero.2 hn.ne' (h <| by simpa using hx)
rwa [zpow_eq_zpow_iff_modEq, orderOf_eq_zero_iff.2 h, Nat.cast_zero, Int.modEq_zero_iff] at hnm
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | injective_zpow_iff_not_isOfFinOrder | null |
Subgroup.zpowers_eq_zpowers_iff {x y : G} (hx : ¬IsOfFinOrder x) :
zpowers x = zpowers y ↔ x = y ∨ x⁻¹ = y := by
refine ⟨fun h ↦ ?_, by rintro (rfl | rfl) <;> simp⟩
have hx_mem : x ∈ zpowers y := by simp [← h]
have hy_mem : y ∈ zpowers x := by simp [h]
obtain ⟨k, rfl⟩ := mem_zpowers_iff.mp hy_mem
obtain ⟨l, hl⟩ := mem_zpowers_iff.mp hx_mem
rw [← zpow_mul] at hl
nth_rewrite 2 [← zpow_one x] at hl
have h1 := (injective_zpow_iff_not_isOfFinOrder.mpr hx) hl
rcases (Int.mul_eq_one_iff_eq_one_or_neg_one).mp h1 with (h | h) <;> simp [h.1] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Subgroup.zpowers_eq_zpowers_iff | null |
@[to_additive]
exists_zpow_eq_one (x : G) : ∃ (i : ℤ) (_ : i ≠ 0), x ^ (i : ℤ) = 1 := by
obtain ⟨w, hw1, hw2⟩ := isOfFinOrder_of_finite x
refine ⟨w, Int.natCast_ne_zero.mpr (_root_.ne_of_gt hw1), ?_⟩
rw [zpow_natCast]
exact (isPeriodicPt_mul_iff_pow_eq_one _).mp hw2
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | exists_zpow_eq_one | null |
mem_powers_iff_mem_zpowers : y ∈ powers x ↔ y ∈ zpowers x :=
(isOfFinOrder_of_finite _).mem_powers_iff_mem_zpowers
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | mem_powers_iff_mem_zpowers | null |
powers_eq_zpowers (x : G) : (powers x : Set G) = zpowers x :=
(isOfFinOrder_of_finite _).powers_eq_zpowers
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | powers_eq_zpowers | null |
mem_zpowers_iff_mem_range_orderOf [DecidableEq G] :
y ∈ zpowers x ↔ y ∈ (Finset.range (orderOf x)).image (x ^ ·) :=
(isOfFinOrder_of_finite _).mem_zpowers_iff_mem_range_orderOf | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | mem_zpowers_iff_mem_range_orderOf | null |
@[to_additive
/-- The equivalence between `Subgroup.zmultiples` of two elements `a, b` of the same additive
order, mapping `i • a` to `i • b`. -/]
noncomputable zpowersEquivZPowers (h : orderOf x = orderOf y) :
Subgroup.zpowers x ≃ Subgroup.zpowers y :=
(finEquivZPowers <| isOfFinOrder_of_finite _).symm.trans <| (finCongr h).trans <|
finEquivZPowers <| isOfFinOrder_of_finite _
@[to_additive (attr := simp) zmultiples_equiv_zmultiples_apply] | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | zpowersEquivZPowers | The equivalence between `Subgroup.zpowers` of two elements `x, y` of the same order, mapping
`x ^ i` to `y ^ i`. |
zpowersEquivZPowers_apply (h : orderOf x = orderOf y) (n : ℕ) :
zpowersEquivZPowers h ⟨x ^ n, n, zpow_natCast x n⟩ = ⟨y ^ n, n, zpow_natCast y n⟩ := by
rw [zpowersEquivZPowers, Equiv.trans_apply, Equiv.trans_apply, finEquivZPowers_symm_apply, ←
Equiv.eq_symm_apply, finEquivZPowers_symm_apply]
simp [h] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | zpowersEquivZPowers_apply | null |
@[to_additive
/-- See `AddSubgroup.closure_toAddSubmonoid_of_isOfFinOrder` for a version with weaker
assumptions. -/]
Subgroup.closure_toSubmonoid_of_finite {s : Set G} :
(closure s).toSubmonoid = Submonoid.closure s :=
closure_toSubmonoid_of_isOfFinOrder <| by simp [isOfFinOrder_of_finite] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Subgroup.closure_toSubmonoid_of_finite | See `Subgroup.closure_toSubmonoid_of_isOfFinOrder` for a version with weaker assumptions. |
@[to_additive /-- See also `Nat.card_zmultiples`. -/]
Fintype.card_zpowers : Fintype.card (zpowers x) = orderOf x :=
(Fintype.card_eq.2 ⟨finEquivZPowers <| isOfFinOrder_of_finite _⟩).symm.trans <|
Fintype.card_fin (orderOf x)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Fintype.card_zpowers | See also `Nat.card_zpowers`. |
card_zpowers_le (a : G) {k : ℕ} (k_pos : k ≠ 0)
(ha : a ^ k = 1) : Fintype.card (Subgroup.zpowers a) ≤ k := by
rw [Fintype.card_zpowers]
apply orderOf_le_of_pow_eq_one k_pos.bot_lt ha
open QuotientGroup
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | card_zpowers_le | null |
orderOf_dvd_card : orderOf x ∣ Fintype.card G := by
use Fintype.card (G ⧸ zpowers x)
rw [← card_zpowers, mul_comm, ← Fintype.card_prod,
← Fintype.card_congr groupEquivQuotientProdSubgroup]
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_dvd_card | null |
orderOf_dvd_natCard {G : Type*} [Group G] (x : G) : orderOf x ∣ Nat.card G := by
obtain h | h := fintypeOrInfinite G
· simp only [Nat.card_eq_fintype_card, orderOf_dvd_card]
· simp only [card_eq_zero_of_infinite, dvd_zero]
@[to_additive]
nonrec lemma Subgroup.orderOf_dvd_natCard {G : Type*} [Group G] (s : Subgroup G) {x} (hx : x ∈ s) :
orderOf x ∣ Nat.card s := by
simpa using orderOf_dvd_natCard (⟨x, hx⟩ : s)
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_dvd_natCard | null |
Subgroup.orderOf_le_card {G : Type*} [Group G] (s : Subgroup G) (hs : (s : Set G).Finite)
{x} (hx : x ∈ s) : orderOf x ≤ Nat.card s :=
le_of_dvd (Nat.card_pos_iff.2 <| ⟨s.coe_nonempty.to_subtype, hs.to_subtype⟩) <|
s.orderOf_dvd_natCard hx
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Subgroup.orderOf_le_card | null |
Submonoid.orderOf_le_card {G : Type*} [Group G] (s : Submonoid G) (hs : (s : Set G).Finite)
{x} (hx : x ∈ s) : orderOf x ≤ Nat.card s := by
rw [← Nat.card_submonoidPowers]; exact Nat.card_mono hs <| powers_le.2 hx
@[to_additive (attr := simp) card_nsmul_eq_zero'] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Submonoid.orderOf_le_card | null |
pow_card_eq_one' {G : Type*} [Group G] {x : G} : x ^ Nat.card G = 1 :=
orderOf_dvd_iff_pow_eq_one.mp <| orderOf_dvd_natCard _
@[to_additive (attr := simp) card_nsmul_eq_zero] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_card_eq_one' | null |
pow_card_eq_one : x ^ Fintype.card G = 1 := by
rw [← Nat.card_eq_fintype_card, pow_card_eq_one']
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_card_eq_one | null |
Subgroup.pow_index_mem {G : Type*} [Group G] (H : Subgroup G) [Normal H] (g : G) :
g ^ index H ∈ H := by rw [← eq_one_iff, QuotientGroup.mk_pow H, index, pow_card_eq_one']
@[to_additive (attr := simp) mod_card_nsmul] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Subgroup.pow_index_mem | null |
pow_mod_card (a : G) (n : ℕ) : a ^ (n % card G) = a ^ n := by
rw [eq_comm, ← pow_mod_orderOf, ← Nat.mod_mod_of_dvd n orderOf_dvd_card, pow_mod_orderOf]
@[to_additive (attr := simp) mod_card_zsmul] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_mod_card | null |
zpow_mod_card (a : G) (n : ℤ) : a ^ (n % Fintype.card G : ℤ) = a ^ n := by
rw [eq_comm, ← zpow_mod_orderOf, ← Int.emod_emod_of_dvd n
(Int.natCast_dvd_natCast.2 orderOf_dvd_card), zpow_mod_orderOf]
@[to_additive (attr := simp) mod_natCard_nsmul] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | zpow_mod_card | null |
pow_mod_natCard {G} [Group G] (a : G) (n : ℕ) : a ^ (n % Nat.card G) = a ^ n := by
rw [eq_comm, ← pow_mod_orderOf, ← Nat.mod_mod_of_dvd n <| orderOf_dvd_natCard _, pow_mod_orderOf]
@[to_additive (attr := simp) mod_natCard_zsmul] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_mod_natCard | null |
zpow_mod_natCard {G} [Group G] (a : G) (n : ℤ) : a ^ (n % Nat.card G : ℤ) = a ^ n := by
rw [eq_comm, ← zpow_mod_orderOf, ← Int.emod_emod_of_dvd n <|
Int.natCast_dvd_natCast.2 <| orderOf_dvd_natCard _, zpow_mod_orderOf] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | zpow_mod_natCard | null |
@[to_additive (attr := simps) /-- If `gcd(|G|,n)=1` then the smul by `n` is a bijection -/]
noncomputable powCoprime {G : Type*} [Group G] (h : (Nat.card G).Coprime n) : G ≃ G where
toFun g := g ^ n
invFun g := g ^ (Nat.card G).gcdB n
left_inv g := by
have key := congr_arg (g ^ ·) ((Nat.card G).gcd_eq_gcd_ab n)
dsimp only at key
rwa [zpow_add, zpow_mul, zpow_mul, zpow_natCast, zpow_natCast, zpow_natCast, h.gcd_eq_one,
pow_one, pow_card_eq_one', one_zpow, one_mul, eq_comm] at key
right_inv g := by
have key := congr_arg (g ^ ·) ((Nat.card G).gcd_eq_gcd_ab n)
dsimp only at key
rwa [zpow_add, zpow_mul, zpow_mul', zpow_natCast, zpow_natCast, zpow_natCast, h.gcd_eq_one,
pow_one, pow_card_eq_one', one_zpow, one_mul, eq_comm] at key
@[to_additive] | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | powCoprime | If `gcd(|G|,n)=1` then the `n`th power map is a bijection |
powCoprime_one {G : Type*} [Group G] (h : (Nat.card G).Coprime n) : powCoprime h 1 = 1 :=
one_pow n
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | powCoprime_one | null |
powCoprime_inv {G : Type*} [Group G] (h : (Nat.card G).Coprime n) {g : G} :
powCoprime h g⁻¹ = (powCoprime h g)⁻¹ :=
inv_pow g n
@[to_additive Nat.Coprime.nsmul_right_bijective] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | powCoprime_inv | null |
Nat.Coprime.pow_left_bijective {G} [Group G] (hn : (Nat.card G).Coprime n) :
Bijective (· ^ n : G → G) :=
(powCoprime hn).bijective
/- TODO: Generalise to `Submonoid.powers`. -/
@[to_additive] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Nat.Coprime.pow_left_bijective | null |
image_range_orderOf [DecidableEq G] :
letI : Fintype (zpowers x) := (Subgroup.zpowers x).instFintypeSubtypeMemOfDecidablePred
Finset.image (fun i => x ^ i) (Finset.range (orderOf x)) = (zpowers x : Set G).toFinset := by
letI : Fintype (zpowers x) := (Subgroup.zpowers x).instFintypeSubtypeMemOfDecidablePred
ext x
rw [Set.mem_toFinset, SetLike.mem_coe, mem_zpowers_iff_mem_range_orderOf]
/- TODO: Generalise to `Finite` + `CancelMonoid`. -/
@[to_additive gcd_nsmul_card_eq_zero_iff] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | image_range_orderOf | null |
pow_gcd_card_eq_one_iff : x ^ n = 1 ↔ x ^ gcd n (Fintype.card G) = 1 :=
⟨fun h => pow_gcd_eq_one _ h <| pow_card_eq_one, fun h => by
let ⟨m, hm⟩ := gcd_dvd_left n (Fintype.card G)
rw [hm, pow_mul, h, one_pow]⟩ | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | pow_gcd_card_eq_one_iff | null |
smul_eq_of_le_smul
{G : Type*} [Group G] [Finite G] {α : Type*} [PartialOrder α] {g : G} {a : α}
[MulAction G α] [CovariantClass G α HSMul.hSMul LE.le] (h : a ≤ g • a) : g • a = a := by
have key := smul_mono_right g (le_pow_smul h (Nat.card G - 1))
rw [smul_smul, ← _root_.pow_succ',
Nat.sub_one_add_one_eq_of_pos Nat.card_pos, pow_card_eq_one', one_smul] at key
exact le_antisymm key h | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | smul_eq_of_le_smul | null |
smul_eq_of_smul_le
{G : Type*} [Group G] [Finite G] {α : Type*} [PartialOrder α] {g : G} {a : α}
[MulAction G α] [CovariantClass G α HSMul.hSMul LE.le] (h : g • a ≤ a) : g • a = a := by
have key := smul_mono_right g (pow_smul_le h (Nat.card G - 1))
rw [smul_smul, ← _root_.pow_succ',
Nat.sub_one_add_one_eq_of_pos Nat.card_pos, pow_card_eq_one', one_smul] at key
exact le_antisymm h key | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | smul_eq_of_smul_le | null |
@[to_additive
/-- A nonempty idempotent subset of a finite cancellative additive monoid is a submonoid -/]
submonoidOfIdempotent {M : Type*} [LeftCancelMonoid M] [Finite M] (S : Set M)
(hS1 : S.Nonempty) (hS2 : S * S = S) : Submonoid M :=
have pow_mem (a : M) (ha : a ∈ S) (n : ℕ) : a ^ (n + 1) ∈ S := by
induction n with
| zero => rwa [zero_add, pow_one]
| succ n ih =>
rw [← hS2, pow_succ]
exact Set.mul_mem_mul ih ha
{ carrier := S
one_mem' := by
obtain ⟨a, ha⟩ := hS1
rw [← pow_orderOf_eq_one a, ← tsub_add_cancel_of_le (succ_le_of_lt (orderOf_pos a))]
exact pow_mem a ha (orderOf a - 1)
mul_mem' := fun ha hb => (congr_arg₂ (· ∈ ·) rfl hS2).mp (Set.mul_mem_mul ha hb) } | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | submonoidOfIdempotent | A nonempty idempotent subset of a finite cancellative monoid is a submonoid |
@[to_additive /-- A nonempty idempotent subset of a finite additive group is a subgroup -/]
subgroupOfIdempotent {G : Type*} [Group G] [Finite G] (S : Set G) (hS1 : S.Nonempty)
(hS2 : S * S = S) : Subgroup G :=
{ submonoidOfIdempotent S hS1 hS2 with
carrier := S
inv_mem' := fun {a} ha => show a⁻¹ ∈ submonoidOfIdempotent S hS1 hS2 by
rw [← one_mul a⁻¹, ← pow_one a, ← pow_orderOf_eq_one a, ← pow_sub a (orderOf_pos a)]
exact pow_mem ha (orderOf a - 1) } | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | subgroupOfIdempotent | A nonempty idempotent subset of a finite group is a subgroup |
@[to_additive (attr := simps!) smulCardAddSubgroup
/-- If `S` is a nonempty subset of a finite additive group `G`, then `|G| • S` is a subgroup -/]
powCardSubgroup {G : Type*} [Group G] [Fintype G] (S : Set G) (hS : S.Nonempty) : Subgroup G :=
have one_mem : (1 : G) ∈ S ^ Fintype.card G := by
obtain ⟨a, ha⟩ := hS
rw [← pow_card_eq_one]
exact Set.pow_mem_pow ha
subgroupOfIdempotent (S ^ Fintype.card G) ⟨1, one_mem⟩ <| by
classical
apply (Set.eq_of_subset_of_card_le (Set.subset_mul_left _ one_mem) (ge_of_eq _)).symm
simp_rw [← pow_add,
Group.card_pow_eq_card_pow_card_univ S (Fintype.card G + Fintype.card G) le_add_self] | def | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | powCardSubgroup | If `S` is a nonempty subset of a finite group `G`, then `S ^ |G|` is a subgroup |
protected IsOfFinOrder.eq_one (ha₀ : 0 ≤ a) (ha : IsOfFinOrder a) : a = 1 := by
obtain ⟨n, hn, ha⟩ := ha.exists_pow_eq_one
exact (pow_eq_one_iff_of_nonneg ha₀ hn.ne').1 ha | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.eq_one | null |
protected IsOfFinOrder.eq_neg_one (ha₀ : a ≤ 0) (ha : IsOfFinOrder a) : a = -1 :=
(sq_eq_one_iff.1 <| ha.pow.eq_one <| sq_nonneg a).resolve_left <| by
rintro rfl; exact one_pos.not_ge ha₀ | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.eq_neg_one | null |
orderOf_abs_ne_one (h : |x| ≠ 1) : orderOf x = 0 := by
rw [orderOf_eq_zero_iff']
intro n hn hx
replace hx : |x| ^ n = 1 := by simpa only [abs_one, abs_pow] using congr_arg abs hx
rcases h.lt_or_gt with h | h
· exact ((pow_lt_one₀ (abs_nonneg x) h hn.ne').ne hx).elim
· exact ((one_lt_pow₀ h hn.ne').ne' hx).elim | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_abs_ne_one | null |
LinearOrderedRing.orderOf_le_two : orderOf x ≤ 2 := by
rcases ne_or_eq |x| 1 with h | h
· simp [orderOf_abs_ne_one h]
rcases eq_or_eq_neg_of_abs_eq h with (rfl | rfl)
· simp
exact orderOf_le_of_pow_eq_one zero_lt_two (by simp) | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | LinearOrderedRing.orderOf_le_two | null |
@[to_additive]
protected Prod.orderOf (x : α × β) : orderOf x = (orderOf x.1).lcm (orderOf x.2) :=
minimalPeriod_prodMap _ _ _
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Prod.orderOf | null |
orderOf_fst_dvd_orderOf : orderOf x.1 ∣ orderOf x :=
minimalPeriod_fst_dvd
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_fst_dvd_orderOf | null |
orderOf_snd_dvd_orderOf : orderOf x.2 ∣ orderOf x :=
minimalPeriod_snd_dvd
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_snd_dvd_orderOf | null |
IsOfFinOrder.fst {x : α × β} (hx : IsOfFinOrder x) : IsOfFinOrder x.1 :=
hx.mono orderOf_fst_dvd_orderOf
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.fst | null |
IsOfFinOrder.snd {x : α × β} (hx : IsOfFinOrder x) : IsOfFinOrder x.2 :=
hx.mono orderOf_snd_dvd_orderOf
@[to_additive IsOfFinAddOrder.prod_mk] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.snd | null |
IsOfFinOrder.prod_mk : IsOfFinOrder a → IsOfFinOrder b → IsOfFinOrder (a, b) := by
simpa only [← orderOf_pos_iff, Prod.orderOf] using Nat.lcm_pos
@[to_additive] | theorem | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | IsOfFinOrder.prod_mk | null |
Prod.orderOf_mk : orderOf (a, b) = Nat.lcm (orderOf a) (orderOf b) :=
(a, b).orderOf | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Prod.orderOf_mk | null |
@[simp]
Nat.cast_card_eq_zero (R) [AddGroupWithOne R] [Fintype R] : (Fintype.card R : R) = 0 := by
rw [← nsmul_one, card_nsmul_eq_zero] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | Nat.cast_card_eq_zero | null |
CharP.addOrderOf_one : CharP R (addOrderOf (1 : R)) where
cast_eq_zero_iff n := by rw [← Nat.smul_one_eq_cast, addOrderOf_dvd_iff_nsmul_eq_zero]
variable [Fintype R]
variable {R} in | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | CharP.addOrderOf_one | null |
charP_of_ne_zero (hn : card R = p) (hR : ∀ i < p, (i : R) = 0 → i = 0) : CharP R p where
cast_eq_zero_iff n := by
have H : (p : R) = 0 := by rw [← hn, Nat.cast_card_eq_zero]
constructor
· intro h
rw [← Nat.mod_add_div n p, Nat.cast_add, Nat.cast_mul, H, zero_mul, add_zero] at h
rw [Nat.dvd_iff_mod_eq_zero]
apply hR _ (Nat.mod_lt _ _) h
rw [← hn, Fintype.card_pos_iff]
exact ⟨0⟩
· rintro ⟨n, rfl⟩
rw [Nat.cast_mul, H, zero_mul] | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | charP_of_ne_zero | null |
charP_of_prime_pow_injective (R) [Ring R] [Fintype R] (p n : ℕ) [hp : Fact p.Prime]
(hn : card R = p ^ n) (hR : ∀ i ≤ n, (p : R) ^ i = 0 → i = n) : CharP R (p ^ n) := by
obtain ⟨c, hc⟩ := CharP.exists R
have hcpn : c ∣ p ^ n := by rw [← CharP.cast_eq_zero_iff R c, ← hn, Nat.cast_card_eq_zero]
obtain ⟨i, hi, rfl⟩ : ∃ i ≤ n, c = p ^ i := by rwa [Nat.dvd_prime_pow hp.1] at hcpn
obtain rfl : i = n := hR i hi <| by rw [← Nat.cast_pow, CharP.cast_eq_zero]
assumption | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | charP_of_prime_pow_injective | null |
@[to_additive]
orderOf_eq [Group G] (a : G) {x y : G} (h : SemiconjBy a x y) : orderOf x = orderOf y := by
rw [orderOf_eq_orderOf_iff]
intro n
exact (h.pow_right n).eq_one_iff | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_eq | null |
orderOf_piMulSingle {ι : Type*} [DecidableEq ι] {M : ι → Type*} [(i : ι) → Monoid (M i)]
(i : ι) (g : M i) :
orderOf (Pi.mulSingle i g) = orderOf g :=
orderOf_injective (MonoidHom.mulSingle M i) (Pi.mulSingle_injective i) g | lemma | GroupTheory | [
"Mathlib.Algebra.CharP.Defs",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.Group.Pointwise.Set.Finite",
"Mathlib.Algebra.Group.Subgroup.Finite",
"Mathlib.Algebra.Module.NatInt",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.ModEq",
"Mathlib.Dyna... | Mathlib/GroupTheory/OrderOfElement.lean | orderOf_piMulSingle | null |
IsPGroup : Prop :=
∀ g : G, ∃ k : ℕ, g ^ p ^ k = 1
variable {p} {G} | def | GroupTheory | [
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.GroupTheory.SpecificGroups.Cyclic"
] | Mathlib/GroupTheory/PGroup.lean | IsPGroup | A p-group is a group in which every element has prime power order |
iff_orderOf [hp : Fact p.Prime] : IsPGroup p G ↔ ∀ g : G, ∃ k : ℕ, orderOf g = p ^ k :=
forall_congr' fun g =>
⟨fun ⟨_, hk⟩ =>
Exists.imp (fun _ h => h.right)
((Nat.dvd_prime_pow hp.out).mp (orderOf_dvd_of_pow_eq_one hk)),
Exists.imp fun k hk => by rw [← hk, pow_orderOf_eq_one]⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.GroupTheory.SpecificGroups.Cyclic"
] | Mathlib/GroupTheory/PGroup.lean | iff_orderOf | null |
of_card {n : ℕ} (hG : Nat.card G = p ^ n) : IsPGroup p G := fun g =>
⟨n, by rw [← hG, pow_card_eq_one']⟩ | theorem | GroupTheory | [
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.GroupTheory.SpecificGroups.Cyclic"
] | Mathlib/GroupTheory/PGroup.lean | of_card | null |
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