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Subgroups of cyclic groups

In abstract algebra, every subgroup of a cyclic group is cyclic. Moreover, for a finite cyclic group of order n, every subgroup's order is a divisor of n, and there is exactly one subgroup for each di

Jordan–Schur theorem

In mathematics, the Jordan–Schur theorem also known as Jordan's theorem on finite linear groups is a theorem in its original form due to Camille Jordan. In that form, it states that there is a functio

Matsumoto's theorem (group theory)

In group theory, Matsumoto's theorem, proved by Hideya Matsumoto, gives conditions for two reduced words of a Coxeter group to represent the same element.

Schur–Zassenhaus theorem

The Schur–Zassenhaus theorem is a theorem in group theory which states that if is a finite group, and is a normal subgroup whose order is coprime to the order of the quotient group , then is a semidir

Krull–Schmidt theorem

In mathematics, the Krull–Schmidt theorem states that a group subjected to certain finiteness conditions on chains of subgroups, can be uniquely written as a finite direct product of indecomposable su

Prüfer theorems

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.

Gromov's theorem on groups of polynomial growth

In geometric group theory, Gromov's theorem on groups of polynomial growth, first proved by Mikhail Gromov, characterizes finitely generated groups of polynomial growth, as those groups which have nil

Brauer–Nesbitt theorem

In mathematics, the Brauer–Nesbitt theorem can refer to several different theorems proved by Richard Brauer and Cecil J. Nesbitt in the representation theory of finite groups. In modular representatio

Maschke's theorem

In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible piece

Hahn embedding theorem

In mathematics, especially in the area of abstract algebra dealing with ordered structures on abelian groups, the Hahn embedding theorem gives a simple description of all linearly ordered abelian grou

Hall–Higman theorem

In mathematical group theory, the Hall–Higman theorem, due to Philip Hall and Graham Higman , describes the possibilities for the minimal polynomial of an element of prime power order for a representa

Frobenius determinant theorem

In mathematics, the Frobenius determinant theorem was a conjecture made in 1896 by the mathematician Richard Dedekind, who wrote a letter to F. G. Frobenius about it (reproduced in, with an English tr

Peter–Weyl theorem

In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved b

Schreier refinement theorem

In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there

Closed-subgroup theorem

In mathematics, the closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of a Lie group G, then H is a

Weil's conjecture on Tamagawa numbers

In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number of a simply connected simple algebraic group defined over a number field is 1. In this case, simply co

Higman–Sims asymptotic formula

In finite group theory, the Higman–Sims asymptotic formula gives an asymptotic estimate on number of groups of prime power order.

Focal subgroup theorem

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

Cartan–Dieudonné theorem

In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an n-dimensional symmetric bilinear space can be described

Stallings theorem about ends of groups

In the mathematical subject of group theory, the Stallings theorem about ends of groups states that a finitely generated group G has more than one end if and only if the group G admits a nontrivial de

Scott core theorem

In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott,. The precise statement is as follows: Given a 3-manifold

Fitting's theorem

Fitting's theorem is a mathematical theorem proved by Hans Fitting. It can be stated as follows: If M and N are nilpotent normal subgroups of a group G, then their product MN is also a nilpotent norma

Ribet's lemma

In mathematics, Ribet's lemma gives conditions for a subgroup of a product of groups to be the whole product group. It was introduced by Ribet .

Fundamental theorem of Galois theory

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his de

Adian–Rabin theorem

In the mathematical subject of group theory, the Adian–Rabin theorem is a result that states that most "reasonable" properties of finitely presentable groups are algorithmically undecidable. The theor

Frobenius's theorem (group theory)

In mathematical group theory, Frobenius's theorem states that if n divides the order of a finite group G, then the number of solutions of xn = 1 is a multiple of n. It was introduced by Frobenius.

Hajós's theorem

In group theory, Hajós's theorem states that if a finite abelian group is expressed as the Cartesian product of simplexes, that is, sets of the form where is the identity element, then at least one of

Nagao's theorem

In mathematics, Nagao's theorem, named after , is a result about the structure of the group of 2-by-2 invertible matrices over the ring of polynomials over a field. It has been extended by Serre to gi

Hurwitz's automorphisms theorem

In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that

Higman's embedding theorem

In group theory, Higman's embedding theorem states that every finitely generated recursively presented group R can be embedded as a subgroup of some finitely presented group G. This is a result of Gra

Grushko theorem

In the mathematical subject of group theory, the Grushko theorem or the Grushko–Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free pro

Golod–Shafarevich theorem

In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem,

Sims conjecture

In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims. He conjectured that if is a primitive permutation group on a finite set and denotes the stabilizer

Universal embedding theorem

The universal embedding theorem, or Krasner–Kaloujnine universal embedding theorem, is a theorem from the mathematical discipline of group theory first published in 1951 by Marc Krasner and Lev Kaluzn

Tits alternative

In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups.

Kurosh subgroup theorem

In the mathematical field of group theory, the Kurosh subgroup theorem describes the algebraic structure of subgroups of free products of groups. The theorem was obtained by Alexander Kurosh, a Russia

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