TY - JOUR

T1 - On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function

AU - Liebeck, Martin W.

AU - Martin, Benjamin M.S.

AU - Shalev, Aner

PY - 2005/6/15

Y1 - 2005/6/15

N2 - We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a long-standing open problem. The proof uses, among other tools, some methods from geometric invariant theory. Using this result, we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of Lie-type groups of fixed rank, drawing conclusions regarding the behaviour of the corresponding "zeta function" ζG(S) = ∑M max G |G : M|-s, which appears in many probabilistic applications. More specifically, we are able to show that for simple groups G and for any fixed real number s > 1, ζG(s) → 0 as |G| → ∞. This confirms a conjecture made in [27, page 84]. We also apply these results to prove the conjecture made in [28, Conjecture 1, page 343], that the symmetric group Sn has n°(1) conjugacy classes of primitive maximal subgroups.

AB - We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a long-standing open problem. The proof uses, among other tools, some methods from geometric invariant theory. Using this result, we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of Lie-type groups of fixed rank, drawing conclusions regarding the behaviour of the corresponding "zeta function" ζG(S) = ∑M max G |G : M|-s, which appears in many probabilistic applications. More specifically, we are able to show that for simple groups G and for any fixed real number s > 1, ζG(s) → 0 as |G| → ∞. This confirms a conjecture made in [27, page 84]. We also apply these results to prove the conjecture made in [28, Conjecture 1, page 343], that the symmetric group Sn has n°(1) conjugacy classes of primitive maximal subgroups.

UR - http://www.scopus.com/inward/record.url?scp=22544466726&partnerID=8YFLogxK

U2 - 10.1215/S0012-7094-04-12834-9

DO - 10.1215/S0012-7094-04-12834-9

M3 - Review article

AN - SCOPUS:22544466726

SN - 0012-7094

VL - 128

SP - 541

EP - 557

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 3

ER -