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23773 | \section{Non-Zero Natural Numbers under Multiplication form Commutative Monoid}
Tags: Natural Numbers, Examples of Monoids, Monoid Examples, Examples of Commutative Monoids
\begin{theorem}
Let $\N_{>0}$ be the set of natural numbers without zero, i.e. $\N_{>0} = \N \setminus \set 0$.
The structure $\struct{\N_{>0}, \t... |
23774 | \section{Non-Zero Rational Numbers Closed under Multiplication}
Tags: Algebraic Closure, Rational Multiplication, Rational Numbers
\begin{theorem}
The set of non-zero rational numbers is closed under multiplication.
\end{theorem}
\begin{proof}
Recall that Rational Numbers form Field under the operations of addition a... |
23775 | \section{Non-Zero Rational Numbers under Multiplication form Infinite Abelian Group}
Tags: Abelian Groups, Examples of Abelian Groups, Group Examples, Abelian Groups: Examples, Infinite Groups: Examples, Rational Multiplication, Examples of Infinite Groups, Abelian Group Examples, Rational Numbers
\begin{theorem}
Let ... |
23776 | \section{Non-Zero Real Numbers Closed under Multiplication}
Tags: Real Numbers, Algebraic Closure, Non-Zero Real Numbers Closed under Multiplication, Real Multiplication
\begin{theorem}
The set of non-zero real numbers is closed under multiplication:
:$\forall x, y \in \R_{\ne 0}: x \times y \in \R_{\ne 0}$
\end{theor... |
23777 | \section{Non-Zero Real Numbers under Multiplication form Abelian Group}
Tags: Non-Zero Real Numbers under Multiplication form Abelian Group, Abelian Groups, Examples of Abelian Groups, Group Examples, Abelian Groups: Examples, Real Numbers, Infinite Groups: Examples, Examples of Infinite Groups, Abelian Group Examples,... |
23778 | \section{Non-Zero Value of Continuous Real-Valued Function has Neighborhood not including Zero}
Tags: Continuous Mappings on Metric Spaces
\begin{theorem}
Let $M = \struct {A, d}$ be a metric space.
Let $f: M \to \R$ be a continuous real-valued function.
Let $\map f a > 0$ for some $a \in M$.
Then there exists $\delta... |
23779 | \section{Nonconstant Periodic Function with no Period is Discontinuous Everywhere}
Tags: Periodic Functions
\begin{theorem}
Let $f$ be a real periodic function that does not have a period.
Then $f$ is either constant or discontinuous everywhere.
\end{theorem}
\begin{proof}
Let $f$ be a real periodic function that doe... |
23780 | \section{Nonempty Grothendieck Universe contains Von Neumann Natural Numbers}
Tags: Set Theory, Grothendieck Universes
\begin{theorem}
Let $\mathbb U$ be a non-empty Grothendieck universe.
Let $\N$ denote the set of von Neumann natural numbers.
Then $\N$ is a subset of $\mathbb U$.
\end{theorem}
\begin{proof}
We prov... |
23781 | \section{Nonlimit Ordinal Cofinal to One}
Tags: Ordinals
\begin{theorem}
Let $x$ be a nonlimit non-empty ordinal.
Let $\operatorname{cof}$ denote the cofinal relation.
Let $1$ denote the ordinal one.
Then:
:$\operatorname{cof} \left({x, 1}\right)$
\end{theorem}
\begin{proof}
Since $1 = 0^+$, $1$ is not a limit ordin... |
23782 | \section{Nonnegative Quadratic Functional implies no Interior Conjugate Points}
Tags: Calculus of Variations
\begin{theorem}
If the quadratic functional
:$\ds \int_a^b \paren {P h'^2 + Q h^2} \rd x$
where:
:$\forall x \in \closedint a b: \map P x > 0$
is nonnegative for all $\map h x$:
:$\map h a = \map h b = 0$
then... |
23783 | \section{Nontrivial Zeroes of Riemann Zeta Function are Symmetrical with respect to Critical Line}
Tags: Riemann Zeta Function
\begin{theorem}
The nontrivial zeroes of the Riemann $\zeta$ function are distributed symmetrically {{WRT}} the critical line.
That is, suppose $s_1 = \sigma_1 + i t$ is a nontrivial zero of ... |
23784 | \section{Nonzero Ideal of Polynomial Ring over Field has Unique Monic Generator}
Tags: Polynomial Rings
\begin{theorem}
Let $K$ be a field.
Let $K \sqbrk x$ be the polynomial ring in one variable over $K$.
Let $I \subseteq K \sqbrk x$ be a nonzero ideal.
Then $I$ is generated by a unique monic polynomial.
\end{theorem... |
23785 | \section{Nonzero natural number is another natural number successor}
Tags: Natural Numbers, Proofs by Induction
\begin{theorem}
Let $\N$ be the 0-based natural numbers:
:$\N = \left\{{0, 1, 2, \ldots}\right\}$
Let $s: \N \to \N: \map s n = n + 1$ be the successor function.
Then:
:$\forall n \in \N \setminus \set 0 \pa... |
23786 | \section{Norm Sequence of Cauchy Sequence has Limit}
Tags: Cauchy Sequences, Normed Division Rings
\begin{theorem}
Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.
Let $\sequence {x_n}$ be a Cauchy sequence in $R$.
Then $\sequence {\norm {x_n} }$ has a limit in $\R$.
That is,
:$\exists l \in \R: \ds ... |
23787 | \section{Norm is Complete Iff Equivalent Norm is Complete}
Tags: Normed Division Rings, Complete Metric Spaces
\begin{theorem}
Let $R$ be a division ring.
Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be equivalent norms on $R$.
Then:
:$\struct {R,\norm {\,\cdot\,}_1}$ is complete {{iff}} $\struct {R,\norm {\,\c... |
23788 | \section{Norm is Continuous}
Tags: Continuity, Norm Theory, Continuous Mappings
\begin{theorem}
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space.
Then the mapping $x \mapsto \norm x$ is continuous.
Here, the metric used is the metric $d$ induced by $\norm {\,\cdot\,}$.
\end{theorem}
\begin{proof}
Since ... |
23789 | \section{Norm of Adjoint}
Tags: Adjoints
\begin{theorem}
Let $H, K$ be Hilbert spaces.
Let $A \in \map B {H, K}$ be a bounded linear transformation.
Then the norm of $A$ satisfies:
:$\norm A^2 = \norm {A^*}^2 = \norm {A^* A}$
where $A^*$ denotes the adjoint of $A$.
\end{theorem}
\begin{proof}
Let $h \in H$ such that ... |
23790 | \section{Norm of Eisenstein Integer}
Tags: Number Theory, Complex Analysis, Algebraic Number Theory
\begin{theorem}
Let $\alpha$ be an Eisenstein integer.
That is, $\alpha = a + b \omega$ for some $a, b \in \Z$, where $\omega = e^{2\pi i /3}$.
Then:
:$\cmod \alpha^2 = a^2 - a b + b^2$
where $\cmod {\, \cdot \,}$ denot... |
23791 | \section{Norm of Hermitian Operator}
Tags: Adjoints, Definitions: Adjoints
\begin{theorem}
Let $\mathbb F \in \set {\R, \C}$.
Let $\HH$ be a Hilbert space over $\mathbb F$.
Let $A : \HH \to \HH$ be a bounded Hermitian operator.
Let $\innerprod \cdot \cdot_\HH$ denote the inner product on $\HH$.
Let $\norm \cdot_\HH$ ... |
23792 | \section{Norm of Unit of Normed Division Algebra}
Tags: Norm Theory, Algebras
\begin{theorem}
Let $\struct {A_F, \oplus}$ be a normed division algebra.
Let the unit of $\struct {A_F, \oplus}$ be $1_A$.
Then:
:$\norm {1_A} = 1$
where $\norm {1_A}$ denotes the norm of $1_A$.
\end{theorem}
\begin{proof}
By definition:
:... |
23793 | \section{Norm of Vector Cross Product}
Tags: Vector Cross Product, Vector Algebra
\begin{theorem}
Let $\mathbf a$ and $\mathbf b$ be vectors in the Euclidean space $\R^3$.
Let $\times$ denote the vector cross product.
Then:
:$(1): \quad$ $\left\Vert{ \mathbf a \times \mathbf b }\right\Vert^2 = \left\Vert{\mathbf a}\ri... |
23794 | \section{Norm on Bounded Linear Functional is Finite}
Tags: Hilbert Spaces
\begin{theorem}
Let $H$ be a Hilbert space.
Let $L$ be a bounded linear functional on $H$.
Let $\norm L$ denote the norm on $L$ defined as:
:$\norm L = \inf \set {c > 0: \forall h \in H: \size {L h} \le c \norm h_H}$
Then:
:$\norm L < \infty$
\... |
23795 | \section{Norm on Bounded Linear Transformation is Finite}
Tags: Hilbert Spaces
\begin{theorem}
Let $H, K$ be Hilbert spaces.
Let $A: H \to K$ be a bounded linear transformation.
Let $\norm A$ denote the norm of $A$ defined by:
:$\norm A = \inf \set {c > 0: \forall h \in H: \norm {A h}_K \le c \norm h_H}$
Then:
:$\norm... |
23796 | \section{Norm on Bounded Linear Transformation is Submultiplicative}
Tags: Linear Transformations on Hilbert Spaces, Bounded Linear Transformations
\begin{theorem}
Let $\struct {X, \norm \cdot_X}$, $\struct {Y, \norm \cdot_Y}$ and $\struct {Z, \norm \cdot_Z}$ be normed vector spaces.
Let $A : X \to Y$ and $B : Y \to Z... |
23797 | \section{Norm on Vector Space is Continuous Function}
Tags: Norm Theory
\begin{theorem}
Let $V$ be a vector space with norm $\norm {\, \cdot \,}$.
The function $\norm {\, \cdot \,}: V \to \R$ is continuous.
\end{theorem}
\begin{proof}
Let $x_n \to x$ in $V$.
We have:
:$x_n \to x \implies \norm {x_n - x} \to 0$
By the... |
23798 | \section{Norm satisfying Parallelogram Law induced by Inner Product}
Tags: Normed Vector Spaces, Inner Product Spaces, Norm satisfying Parallelogram Law induced by Inner Product
\begin{theorem}
Let $V$ be a vector space over $\R$.
Let $\norm \cdot : V \to \R$ be a norm on $V$ such that:
:$\norm {x + y}^2 + \norm {x -... |
23799 | \section{Normal Space is Preserved under Homeomorphism}
Tags: Separation Axioms, Normal Spaces
\begin{theorem}
Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces.
Let $\phi: T_A \to T_B$ be a homeomorphism.
If $T_A$ is a normal space, then so is $T_B$.
\end{theorem}
\begin{proof... |
23800 | \section{Nicely Normed Cayley-Dickson Construction from Associative Algebra is Alternative}
Tags: Cayley-Dickson Construction
\begin{theorem}
Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra.
Let $A' = \left({A_F, \oplus'}\right)$ be constructed from $A$ using the Cayley-Dickson construction.
Then $A'$ is a nicel... |
23801 | \section{Niemytzki Plane is Topology}
Tags: Niemytzki Plane
\begin{theorem}
Niemytzki plane is a topological space.
\end{theorem}
\begin{proof}
By definition $T = \struct {S, \tau}$ is the Niemytzki plane {{iff}}:
{{begin-eqn}}
{{eqn | n = 1
| l = S
| r = \set {\tuple {x, y}: y \ge 0}
}}
{{eqn | n = 2
... |
23802 | \section{Nilpotent Element is Zero Divisor}
Tags: Nilpotent Ring Elements, Zero Divisors, Ring Theory
\begin{theorem}
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.
Suppose further that $R$ is not the null ring.
Let $x \in R$ be a nilpotent element of $R$.
Then $x$ is a zero divisor in $R$.
\end{theorem}
... |
23803 | \section{Nilpotent Elements of Commutative Ring form Ideal}
Tags: Nilpotent Ring Elements, Ring Theory, Ideal Theory, Commutative Rings
\begin{theorem}
Let $\struct {R, +, \circ}$ be a commutative ring whose zero is $0_R$ and whose unity is $1_R$.
The subset of nilpotent elements of $R$ form an ideal of $R$.
\end{theo... |
23804 | \section{Nilpotent Ring Element plus Unity is Unit}
Tags: Nilpotence
\begin{theorem}
Let $A$ be a ring with unity.
Let $1 \in A$ be its unity.
Let $a \in A$ be nilpotent.
Then $1 + a$ is a unit of $A$.
\end{theorem}
\begin{proof}
Because $a$ is nilpotent, there exists a natural number $n > 0$ with $a^n = 0$.
By Sum o... |
23805 | \section{Nine Regular Polyhedra}
Tags: Regular Polyhedra
\begin{theorem}
There exist $9$ regular polyhedra.
\end{theorem}
\begin{proof}
From Five Platonic Solids, there exist $5$ regular polyhedra which are convex:
:the regular tetrahedron
:the cube
:the regular octahedron
:the regular dodecahedron
:the regular icosa... |
23806 | \section{Niven's Theorem}
Tags: Trigonometry, Niven's Theorem
\begin{theorem}
Consider the angles $\theta$ in the range $0 \le \theta \le \dfrac \pi 2$.
The only values of $\theta$ such that both $\dfrac \theta \pi$ and $\sin \theta$ are rational are:
:$\theta = 0: \sin \theta = 0$
:$\theta = \dfrac \pi 6: \sin \theta... |
23807 | \section{Niven's Theorem/Lemma}
Tags: Proofs by Induction, Niven's Theorem
\begin{theorem}
For any integer $n \ge 1$, there exists a polynomial $\map {F_n} x$ such that:
:$\map {F_n} {2 \cos t} = 2 \cos n t$
In addition:
:$\deg F_n = n$
and $F_n$ is a monic polynomial with integer coefficients.
\end{theorem}
\begin{p... |
23808 | \section{No 4 Fibonacci Numbers can be in Arithmetic Sequence}
Tags: Arithmetic Progressions, Arithmetic Sequences, Fibonacci Numbers
\begin{theorem}
Let $a, b, c, d$ be distinct Fibonacci numbers.
Then, except for the trivial case:
:$a = 0, b = 1, c = 2, d = 3$
it is not possible that $a, b, c, d$ are in arithmetic s... |
23809 | \section{No Bijection between Finite Set and Proper Subset}
Tags: Set Theory, No Bijection between Finite Set and Proper Subset, Subsets, Subset
\begin{theorem}
A finite set can not be in one-to-one correspondence with one of its proper subsets.
That is, a finite set is not Dedekind-infinite.
\end{theorem}
\begin{pro... |
23810 | \section{No Bijection from Set to its Power Set}
Tags: Bijections, Power Set, Mappings
\begin{theorem}
Let $S$ be a set.
Let $\powerset S$ denote the power set of $S$.
There is no bijection $f: S \to \powerset S$.
\end{theorem}
\begin{proof}
A bijection is by its definition also a surjection.
By Cantor's Theorem ther... |
23811 | \section{No Boolean Interpretation Models a WFF and its Negation}
Tags: Propositional Logic, Propositional Calculus
\begin{theorem}
Let $v$ be a boolean interpretation.
Let $\mathbf A$ be a WFF of propositional logic.
Then $v$ can not model both $\mathbf A$ and $\neg \mathbf A$.
\end{theorem}
\begin{proof}
Suppose th... |
23812 | \section{No Group has Two Order 2 Elements}
Tags: Group Theory, Order of Group Elements
\begin{theorem}
A group can not contain exactly two elements of order $2$.
\end{theorem}
\begin{proof}
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Suppose:
: $s, t \in \struct {G, \circ}\: s \ne t, \order s = \order... |
23813 | \section{No Infinitely Descending Membership Chains}
Tags: Set Theory
\begin{theorem}
Let $\omega$ denote the minimal infinite successor set.
Let $F$ be a function whose domain is $\omega$.
Then:
:$\exists n \in \omega: \map F {n^+} \notin \map F n$
\end{theorem}
\begin{proof}
Let $F$ be a function whose domain is $\... |
23814 | \section{No Infinitely Descending Membership Chains/Corollary}
Tags: Axiom of Foundation
\begin{theorem}
There cannot exist a sequence $\sequence {x_n}$ whose domain is $\N_{\gt 0}$ such that:
:$\forall n \in \N_{\gt 0}: x_{n+1} \in x_n$
\end{theorem}
\begin{proof}
{{AimForCont}} there is a sequence like that.
From t... |
23815 | \section{No Injection from Power Set to Set}
Tags: Power Set, No Injection from Power Set to Set, Injections
\begin{theorem}
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then there is no injection from $\powerset S$ into $S$.
\end{theorem}
\begin{proof}
The identity mapping $f: \mathcal P(S) \to \math... |
23816 | \section{No Injection from Power Set to Set/Lemma}
Tags: Power Set, No Injection from Power Set to Set
\begin{theorem}
Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
Then there does not exist a set $B$ such that there is an injection from $B$ into $S$ and a surjection from $B$ onto $\powerset S$.
\end{th... |
23817 | \section{No Isomorphism from Woset to Initial Segment}
Tags: Order Isomorphisms, Well-Orderings, Orderings, Order Morphisms
\begin{theorem}
Let $\struct {S, \preceq}$ be a woset.
Let $a \in S$, and let $S_a$ be the initial segment of $S$ determined by $a$.
Then there is no order isomorphism between $S$ and $S_a$.
\end... |
23818 | \section{No Largest Ordinal}
Tags: Ordinals
\begin{theorem}
Let $a$ be a set of ordinals.
Then:
:$\forall x \in a: x \prec \paren {\bigcup a}^+$
\end{theorem}
\begin{proof}
For this proof, we shall use $\prec$, $\in$, and $\subset$ interchangeably.
We are justified in doing this because of Ordering on Ordinal is Subs... |
23819 | \section{No Membership Loops}
Tags: Set Theory, Class Theory, Axiom of Foundation, Axiom of Regularity
\begin{theorem}
For any proper classes or sets $A_1, A_2, \ldots, A_n$:
:$\neg \paren {A_1 \in A_2 \land A_2 \in A_3 \land \cdots \land A_n \in A_1}$
\end{theorem}
\begin{proof}
{{NotZFC}}
Either $A_1, A_2, \ldots, ... |
23820 | \section{No Natural Number between Number and Successor}
Tags: Ordinals, No Ordinal Between Set and Successor, Ordering on Natural Numbers, No Natural Number between Number and Successor
\begin{theorem}
Let $\N$ be the natural numbers.
Let $n \in \N$.
Then no natural number $m$ exists strictly between $n$ and its succ... |
23821 | \section{No Non-Trivial Norm on Rational Numbers is Complete}
Tags: Normed Division Rings, Complete Metric Spaces
\begin{theorem}
No non-trivial norm on the set of the rational numbers is complete.
\end{theorem}
\begin{proof}
By P-adic Norm not Complete on Rational Numbers, no $p$-adic norm $\norm{\,\cdot\,}_p$ on th... |
23822 | \section{No Order Isomophism Between Distinct Initial Segments of Woset}
Tags: Well-Orderings
\begin{theorem}
Let $E$ be a well-ordered set.
Let $S_\alpha, S_\beta$ be initial segments of $E$ that are order isomorphic.
Then $S_\alpha = S_\beta$.
\end{theorem}
\begin{proof}
{{AimForCont}} $S_\alpha \ne S_\beta$.
Then ... |
23823 | \section{No Quadruple of Consecutive Sums of Squares Exists}
Tags: Sums of Squares
\begin{theorem}
It is not possible for a quadruple of consecutive positive integers each of which is the sum of two squares.
\end{theorem}
\begin{proof}
$4$ consecutive positive integers will be in the forms:
{{begin-eqn}}
{{eqn | l = ... |
23824 | \section{No Simple Graph is Perfect}
Tags: Simple Graphs, Graph Theory, Perfect Graphs
\begin{theorem}
Let $G$ be a simple graph whose order is $2$ or greater.
Then $G$ is not perfect.
\end{theorem}
\begin{proof}
Recall that a perfect graph is one where each vertex is of different degree.
We note in passing that the ... |
23825 | \section{No Valid Categorical Syllogism contains two Particular Premises}
Tags: Categorical Syllogisms
\begin{theorem}
Let $Q$ be a valid categorical syllogism.
Then at least one of the premises of $Q$ is universal.
\end{theorem}
\begin{proof}
Suppose both premises of $Q$ are particular.
Then the pattern of $Q$ is on... |
23826 | \section{No Valid Categorical Syllogism with Particular Premise has Universal Conclusion}
Tags: Categorical Syllogisms
\begin{theorem}
Let $Q$ be a valid categorical syllogism.
Let one of the premises of $Q$ be particular.
Then the conclusion of $Q$ is also particular.
\end{theorem}
\begin{proof}
Let the major premis... |
23827 | \section{Noether's Theorem (Calculus of Variations)}
Tags: Physics, Partial Differential Equations, Calculus of Variations
\begin{theorem}
Let $y_i$, $F$, $\Psi_i$, $\Phi$ be real functions.
Let $x, \epsilon \in \R$.
Let $\mathbf y = \sequence {y_i}_{1 \mathop \le i \mathop \le n}$ and $\mathbf \Psi = \sequence{\Psi_i... |
23828 | \section{Noether's Theorem (Hamiltonian Mechanics)}
Tags: Physics, Partial Differential Equations, Calculus of Variations, Hamiltonian Mechanics
\begin{theorem}
Let there be an infinitesimal transformation of generalised coordinates such that:
:$q_i \to \tilde q_i = q_i + q_i^\alpha \tuple{q, \dot q, t} \varepsilon_\a... |
23829 | \section{Noether Normalization Lemma}
Tags: Commutative Algebra, Named Theorems
\begin{theorem}
Let $k$ be a field.
Let $A$ be a non-trivial finitely generated $k$-algebra.
{{explain|the above link is for Definition:Non-Trivial Ring -- we need to define a Definition:Non-Trivial Algebra}}
Then there exists $n \in \N$ a... |
23830 | \section{Noetherian Domain is Factorization Domain}
Tags: Factorization, Ring Theory
\begin{theorem}
Let $R$ be a noetherian integral domain.
Then $R$ is a factorization domain.
\end{theorem}
\begin{proof}
Let $\FF$ be the set of ideals of $R$ of the form $x R$, with $x$ not a unit and such that $x$ cannot be decompo... |
23831 | \section{Noetherian Space is Compact}
Tags: Noetherian Spaces
\begin{theorem}
Let $\struct {X, \tau}$ be a Noetherian topological space.
Then $\struct {X, \tau}$ is compact.
\end{theorem}
\begin{proof}
{{tidy}}
Let $\family {U_i}_{i \mathop \in I}$ be a cover of $X$.
That is, $\bigcup_{i \mathop \in I} U_i = X$.
Let ... |
23832 | \section{Non-Abelian Group of Order p Cubed has Exactly One Normal Subgroup of Order p}
Tags: P-Groups
\begin{theorem}
Let $p$ be a prime number.
Let $G$ be a non-abelian group of order $p^3$.
Then $G$ contains exactly one normal subgroup of order $p$.
\end{theorem}
\begin{proof}
From Center of Group of Prime Power O... |
23833 | \section{Non-Abelian Order 10 Group has Order 5 Element}
Tags: Non-Abelian Order 10 Group has Order 5 Element, Groups of Order 10, Order of Group Elements
\begin{theorem}
Let $G$ be a non-abelian group of order $10$.
Then $G$ has at least one element of order $5$.
\end{theorem}
\begin{proof}
By Lagrange's Theorem, al... |
23834 | \section{Non-Abelian Order 2p Group has Order p Element}
Tags: Order of Group Elements, Groups of Order 2 p
\begin{theorem}
Let $p$ be an odd prime.
Let $G$ be a non-abelian group of order $2 p$.
Then $G$ has at least one element of order $p$.
\end{theorem}
\begin{proof}
By Lagrange's Theorem, all the elements of $G$... |
23835 | \section{Non-Abelian Order 8 Group has Order 4 Element}
Tags: Groups of Order 8
\begin{theorem}
Let $G$ be a non-abelian group of order $8$.
Then $G$ has at least one element of order $4$.
\end{theorem}
\begin{proof}
Let $e \in G$ be the identity of $G$.
Let $g \in G$ be an arbitrary element of $G$ such that $g \ne e... |
23836 | \section{Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group}
Tags: Quaternion Group, Non-Abelian Order 8 Group with One Order 2 Element is Quaternion Group, Groups of Order 8
\begin{theorem}
Let $G$ be a group with the following properties:
:$(1): \quad G$ is non-abelian.
:$(2): \quad G$ is of orde... |
23837 | \section{Non-Abelian Simple Finite Groups are Infinitely Many}
Tags: Simple Groups
\begin{theorem}
There exist infinitely many types of group which are non-abelian and finite.
\end{theorem}
\begin{proof}
We have that Alternating Group is Simple except on 4 Letters.
So for all $n \in \N$ such that $n \ne 4$, the alter... |
23838 | \section{Non-Archimedean Division Ring Iff Non-Archimedean Completion}
Tags: Complete Metric Spaces, Definitions: Norm Theory, Division Rings, Normed Division Rings, Definitions: Division Rings, Norm Theory, Definitions: Complete Metric Spaces, Non-Archimedean Norms
\begin{theorem}
Let $\struct {R, \norm {\, \cdot \,}... |
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