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title: "Z* theorem" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["theorems-about-finite-groups"] topic_path: "general/theorems-about-finite-groups" source: "https://en.wikipedia.org/wiki/Z*_theorem" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, George Glauberman's Z theorem* is stated as follows:

::quote 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 Its theorem 4 states:

::quote 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 | last = Dade | first = Everett C. | year = 1971 | author-link = Everett C. Dade | title = Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969 | editor1-last = Powell | editor1-first = M. B. | editor2-last = Higman | editor2-first = Graham | editor2-link = Graham Higman | publisher = Academic Press | location = Boston, MA | pages = 249–327 | isbn = 978-0-12-563850-0 | mr = 0360785
• {{Citation| title = Central elements in core-free groups | last = Glauberman | first = George | year = 1966 | author-link = George Glauberman | journal = Journal of Algebra | volume = 4 | issue = 3 | pages = 403–420 | doi = 10.1016/0021-8693(66)90030-5 | issn = 0021-8693 | mr = 0202822 | zbl = 0145.02802 | doi-access = free
• {{Citation| title = On extensions of the Baer-Suzuki theorem | last1 = Guralnick | first1 = Robert M. | last2 = Robinson | first2 = Geoffrey R. | journal = Israel Journal of Mathematics | year = 1993 | volume = 82 | issue = 1 | pages = 281–297 | doi = 10.1007/BF02808114 | issn = 0021-2172 | mr = 1239051 | zbl = 0794.20029 | doi-access =
• {{cite journal | title = Centralizers of normal subgroups and the Z*-theorem | last1 = Henke | first1 = Ellen | last2 = Semeraro | first2 = Jason | journal = Journal of Algebra | date = 1 October 2015 | volume = 439 | pages = 511–514 | arxiv = 1411.1932 | doi = 10.1016/j.jalgebra.2015.06.027 | doi-access = free
• {{Citation| title = The Z*-theorem for compact Lie groups | last1 = Mislin | first1 = Guido | last2 = Thévenaz | first2 = Jacques | journal = Mathematische Annalen | year = 1991 | volume = 291 | issue = 1 | pages = 103–111 | url = http://infoscience.epfl.ch/record/130435 | doi = 10.1007/BF01445193 | issn = 0025-5831 | mr = 1125010 | doi-access = free

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