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 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:

  1. g normalizes both U and the centralizer CT(U), that is g is contained in N = NG(U) ∩ NG(CT(U))
  2. t is contained in U and tg ≠ gt
  3. U is generated by the N-conjugates of t
  4. 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 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 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.

Works cited

  • Dade, Everett C. (1971), "Character theory pertaining to finite simple groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969, Boston, MA: Academic Press, pp. 249–327, ISBN 978-0-12-563850-0, MR 0360785 gives a detailed proof of the Brauer–Suzuki theorem.
  • Glauberman, George (1966), "Central elements in core-free groups", Journal of Algebra, 4 (3): 403–420, doi:10.1016/0021-8693(66)90030-5, ISSN 0021-8693, MR 0202822, Zbl 0145.02802
  • Guralnick, Robert M.; Robinson, Geoffrey R. (1993), "On extensions of the Baer-Suzuki theorem", Israel Journal of Mathematics, 82 (1): 281–297, doi:10.1007/BF02808114, ISSN 0021-2172, MR 1239051, Zbl 0794.20029
  • Henke, Ellen; Semeraro, Jason (1 October 2015). "Centralizers of normal subgroups and the Z*-theorem". Journal of Algebra. 439: 511–514. arXiv:1411.1932. doi:10.1016/j.jalgebra.2015.06.027.
  • Mislin, Guido; Thévenaz, Jacques (1991), "The Z*-theorem for compact Lie groups", Mathematische Annalen, 291 (1): 103–111, doi:10.1007/BF01445193, ISSN 0025-5831, MR 1125010