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).
Details
The original paper {{harvtxt|Glauberman|1966}} gave several criteria for an element to lie outside {{nobreak|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:
- 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 {{harvtxt|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 {{harvtxt|Mislin|Thévenaz|1991}}, which also contains several useful results in the finite case.
{{harvtxt|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.
Works cited
{{refbegin}}
- {{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
}} gives a detailed proof of the Brauer–Suzuki theorem.
- {{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
}}
{{refend}}
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