Z* theorem

In mathematics, George Glauberman's Z* theorem is stated as follows:

Z* theorem: Let G be a finite group, with O being its maximal normal subgroup of odd order. If T is a Sylow 2-subgroup of G containing an involution not conjugate in G to any other element of T, then the involution lies in Z*, which is the inverse image in G of the center of G/''O''.

This generalizes the Brauer–Suzuki theorem.

Details

The original paper gave several criteria for an element to lie outside Its theorem 4 states:

For an element t in T, it is necessary and sufficient for t to lie outside Z* that there is some g in G and abelian subgroup U of T satisfying the following properties:
g normalizes both U and the centralizer C_T, that is g is contained in N = N_G ∩ N_G
t is contained in U and tg ≠ gt
U is generated by the N-conjugates of t
the exponent of U is equal to the order of t
Moreover g may be chosen to have prime power order if t is in the center of T, and g may be chosen in T otherwise.

A simple corollary is that an element t in T is not in Z* if and only if there is some s ≠ t such that s and t commute and s and t are G-conjugate.
A generalization to odd primes was recorded in : if t is an element of prime order p and the commutator has order coprime to p for all g, then t is central modulo the p′-core. This was also generalized to odd primes and to compact [Lie group]s in, which also contains several useful results in the finite case.
have also studied an extension of the Z* theorem to pairs of groups with H a normal subgroup of G.

Works cited

gives a detailed proof of the Brauer–Suzuki theorem.

Category:Theorems about finite groups