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Theorems that help decompose a finite group based on prime factors of its order
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fix... | 27233 | abstract_algebra |
Group of even permutations of a finite set
In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by A"n" or Alt("n").
Basic properties.
For "n" &g... | 25195 | abstract_algebra |
Type of group in abstract algebra
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group $\mathrm{S}_n$ defined over a finite set of $n$ ... | 14315 | abstract_algebra |
Cardinality of a mathematical group, or of the subgroup generated by an element
In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is "infinite". The "order" of an element of a group (also called period length or period) is the order of the subgr... | 199412 | abstract_algebra |
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements.
Formall... | 1836466 | abstract_algebra |
Group in which the order of every element is a power of p
In mathematics, specifically group theory, given a prime number "p", a "p"-group is a group in which the order of every element is a power of "p". That is, for each element "g" of a "p"-group "G", there exists a nonnegative integer "n" such that the product of "... | 12162 | abstract_algebra |
Mathematical group that can be generated as the set of powers of a single element
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C"n", that is generated by a single element. That is, it is a set of invertible elements with a single associative b... | 26391 | abstract_algebra |
The order of a subgroup of a finite group G divides the order of G
In the mathematical field of group theory, Lagrange's theorem is a theorem that states that for any finite group G, the order (number of elements) of every subgroup of G divides the order of G. The theorem is named after Joseph-Louis Lagrange. The follo... | 15371 | abstract_algebra |
Representation of groups by permutations
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group.
More specifically, G is isomorphic to a subgroup of the symmetric group $\operatorname{Sym}(G)$ whose elements are the permutations o... | 50326 | abstract_algebra |
Group that can be constructed from abelian groups using extensions
In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the t... | 49669 | abstract_algebra |
Commutative group where every element is the sum of elements from one finite subset
In abstract algebra, an abelian group $(G,+)$ is called finitely generated if there exist finitely many elements $x_1,\dots,x_s$ in $G$ such that every $x$ in $G$ can be written in the form $x = n_1x_1 + n_2x_2 + \cdots + n_sx_s$ for so... | 29251 | abstract_algebra |
In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov.
Statement.
Let "A" be an abelian group. If "A" is finitely generated then by the fundamental theorem of finitely generated abelian groups, "A" is decomp... | 3117305 | abstract_algebra |
In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term "quasinormal subgroup" was introduced by Øystein Ore in 1937.
Two subgroups are said to permute (or... | 667154 | abstract_algebra |
Existence of group elements of prime order
In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive in... | 677473 | abstract_algebra |
In abstract algebra, the focal subgroup theorem describes the fusion of elements in a Sylow subgroup of a finite group. The focal subgroup theorem was introduced in and is the "first major application of the transfer" according to . The focal subgroup theorem relates the ideas of transfer and fusion such as described i... | 2947865 | abstract_algebra |
Classification theorem in group theory
In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson (1962, 1963).
History.
The contrast that these results show between groups of odd and even order suggest... | 461822 | abstract_algebra |
In the area of modern algebra known as group theory, the Mathieu group "M22" is a sporadic simple group of order
27 · 32 · 5 · 7 · 11 = 443520
≈ 4×105.
History and properties.
"M22" is one of the 26 sporadic groups and was introduced by Mathieu (1861, 1873). It is a 3-fold transitive permutation group on 22 objects... | 1961349 | abstract_algebra |
Group of units of the ring of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to "n" from the set $\{0,1,\dots,n-1\}$ of "n" non-negative integers form a group under multiplication modulo "n", called the multiplicative group of integers modulo "n". Equivalently, the elements of this gro... | 367647 | abstract_algebra |
Subgroup mapped to itself under every automorphism of the parent group
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism, e... | 3273 | abstract_algebra |
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