# Category: Theorems in group theory

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
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