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 containing an involution not conjugate in G to any other element of T, then the involution lies in Z*(G), which is the inverse image in G of the center of G/O(G).

This generalizes the Brauer–Suzuki theorem (and the proof uses the Brauer-Suzuki theorem to deal with some small cases).

Details
The original paper  gave several criteria for an element to lie outside Z*(G).  Its theorem 4 states:

For an element t in T, it is necessary and sufficient for t to lie outside Z*(G) that there is some g in G and abelian subgroup U of T satisfying the following properties:
 g normalizes both U and the centralizer CT(U), that is g is contained in N = NG(U) ∩ NG(CT(U))
 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*(G) 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 [t, g] 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 groups in , which also contains several useful results in the finite case.

 have also studied an extension of the Z* theorem to pairs of groups (G,H) with H a normal subgroup of G.

References
 gives a detailed proof of the Brauer–Suzuki theorem.
 
 
 

Category:Finite groups
Category:Theorems in group theory