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).

The original paper (Glauberman 1966) 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:

1. g normalizes both U and the centralizer C_T(U), that is g is contained in N = N_G(U) ∩ N_G(C_T(U))
2. t is contained in U and tg ≠ gt
3. U is generated by the N-conjugates of t
4. 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 (Guralnick & Robinson 1993): 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 (Mislin & Thévenaz 1991), which also contains several useful results in the finite case.

(Henke & Semeraro 2014) 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.