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).

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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*(G) that there is some g in G and abelian subgroup U of T satisfying the following properties: 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.