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*(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.

Works cited

• {{Citation| chapter = Character theory pertaining to finite simple groups | author-link = Everett C. Dade | editor1-last = Powell | editor1-first = M. B. | editor2-last = Higman | editor2-first = Graham | editor2-link = Graham Higman
• {{Citation| title = Central elements in core-free groups | author-link = George Glauberman | doi-access = free
• {{Citation| title = On extensions of the Baer-Suzuki theorem | doi-access =
• {{cite journal | title = Centralizers of normal subgroups and the Z*-theorem | doi-access = free
• {{Citation| title = The Z*-theorem for compact Lie groups | doi-access = free