Resent-From: Daniela NikolovaResent-To: group-pub-forum@maths.bath.ac.uk Date: Mon, 23 Sep 96 12:44:31 BG From: Daniela Nikolova Dear Markku, In your e-mail from September,18 you asked whether a finite simple gp G with a maximal subgroup of order 2p (p- an odd prime) is known to be either PSL(2,q), or Sz(q) for suitable q. In his paper, entitled "Finite nonsolvable groups having a maximal subgroup of order 2p" (PLISKA, Studia mathematica bulgarica. Vol.2, 1981, p. 157-161) Kerope Tchakarian proved that if the order of G is divisible by at most 4 distinct pprimes, then G is isomorphic to PSL(2,q) or Sz(2^q) for an appropriate value of q. That result was proved using group theoretic, character theoretic, and elementary arithmetic arguments. I am pretty sure that after the Classification it is proved elsewhere too. I shall be happy to hear from you again! Best regards, Daniela Nikolova. E-mail: nikolova@bgearn.acad.bg ------------ Date: Thu, 10 Oct 1996 16:07:26 +0100 From: "Robert A. Wilson" Subject: Problem 20. Assuming CFSG, we have the following: 1. It is an easy exercise to show that alternating groups A_n do not have maximal subgroups of order 2p, for n > 5. 2. Given the character tables of the sporadic groups, it is immediate that no subgroup of order 2p is maximal, as it must be contained in a larger Frobenius group. 3. For classical groups, every maximal subgroup is either non-abelian almost simple, or in one of a few geometrically 'obvious' families. A straightforward case-by-case analysis shows that the only ones of order 2p are certain groups of type L1(q^2).2 and U1(q^2).2 in L2(q). 4. For exceptional groups of Lie type, someone who has the theory of tori at her fingertips will instantly supply the argument. Alternatively, the classification of all local maximal subgroups (not maximal local subgroups!) by AM Cohen, MW Liebeck, J Saxl and GM Seitz provides a suitable sledge-hammer with which to crack this nut. (I can't find a reference at the moment, but try Martin Liebeck's survey paper in the Arcata conference proceedings.) Rob Wilson.