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:
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 (2015) have also studied an extension of the Z* theorem to pairs of groups (G, H) with H a normal subgroup of G.