# Category: Theorems about finite groups

Cauchy's theorem (group theory)
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
Trichotomy theorem
In group theory, the trichotomy theorem divides the finite simple groups of characteristic 2 type and rank at least 3 into three classes. It was proved by Aschbacher for rank 3 and by for rank at leas
Gilman–Griess theorem
In finite group theory, a mathematical discipline, the Gilman–Griess theorem, proved by, classifies the finite simple groups of characteristic 2 type with e(G) ≥ 4 that have a "standard component", wh
Baer–Suzuki theorem
In mathematical finite group theory, the Baer–Suzuki theorem, proved by and , states that if any two elements of a conjugacy class C of a finite group generate a nilpotent subgroup, then all elements
Walter theorem
In mathematics, the Walter theorem, proved by John H. Walter , describes the finite groups whose Sylow 2-subgroup is abelian. used Bender's method to give a simpler proof.
Brauer–Suzuki theorem
In mathematics, the Brauer–Suzuki theorem, proved by , , , states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group
Chevalley–Shephard–Todd theorem
In mathematics, the Chevalley–Shephard–Todd theorem in invariant theory of finite groups states that the ring of invariants of a finite group acting on a complex vector space is a polynomial ring if a
B-theorem
B-theorem is a mathematical finite group theory result formerly known as the B-conjecture. The theorem states that if is the centralizer of an involution of a finite group, then every component of is
Burnside's theorem
In mathematics, Burnside's theorem in group theory states that if G is a finite group of order where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence eachnon
Sylow theorems
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 informat
Brauer–Fowler theorem
In mathematical finite group theory, the Brauer–Fowler theorem, proved by , states that if a group G has even order g > 2 then it has a proper subgroup of order greater than g1/3. The technique of the
Thompson transitivity theorem
In mathematical finite group theory, the Thompson transitivity theorem gives conditions under which the centralizer of an abelian subgroup A acts transitively on certain subgroups normalized by A. It
In mathematical finite group theory, the Gorenstein–Harada theorem, proved by Gorenstein and Harada in a 464-page paper, classifies the simple finite groups of sectional 2-rank at most 4. It is part o
Replacement theorem
In mathematical group theory, the Thompson replacement theorem is a theorem about the existence of certain abelian subgroups of a p-group. The Glauberman replacement theorem is a generalization of it
Balance theorem
In mathematical group theory, the balance theorem states that if G is a group with no core then G either has disconnected Sylow 2-subgroups or it is of characteristic 2 type or it is of component type
Alperin–Brauer–Gorenstein theorem
In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups. These are isomorphic either to three-dimensional projecti
Brauer–Suzuki–Wall theorem
In mathematics, the Brauer–Suzuki–Wall theorem, proved by , characterizes the one-dimensional unimodular projective groups over finite fields.
Lagrange's theorem (group theory)
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 th
Z* theorem
In mathematics, George Glauberman's Z* theorem is stated as follows: Z* theorem: Let G be a finite group, with O(G) being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G cont
Cayley's theorem
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
Schreier conjecture
In finite group theory, the Schreier conjecture asserts that the outer automorphism group of every finite simple group is solvable. It was proposed by Otto Schreier in 1926, and is now known to be tru
Feit–Thompson theorem
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 .
Jordan's theorem (symmetric group)
In finite group theory, Jordan's theorem states that if a primitive permutation group G is a subgroup of the symmetric group Sn and contains a p-cycle for some prime number p < n − 2, then G is either
Classical involution theorem
In mathematical finite group theory, the classical involution theorem of Aschbacher classifies simple groups with a classical involution and satisfying some other conditions, showing that they are mos
Component theorem
In the mathematical classification of finite simple groups, the component theorem of Aschbacher shows that if G is a simple group of odd type, and various other assumptions are satisfied, then G has a
L-balance theorem
In mathematical finite group theory, the L-balance theorem was proved by .The letter L stands for the layer of a group, and "balance" refers to the property discussed below.
Thompson uniqueness theorem
In mathematical finite group theory, Thompson's original uniqueness theorem states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary a
ZJ theorem
In mathematics, George Glauberman's ZJ theorem states that if a finite group G is p-constrained and p-stable and has a normal p-subgroup for some odd prime p, then Op′(G)Z(J(S)) is a normal subgroup o
Gorenstein–Walter theorem
In mathematics, the Gorenstein–Walter theorem, proved by Gorenstein and Walter , states that if a finite group G has a dihedral Sylow 2-subgroup, and O(G) is the maximal normal subgroup of odd order,