Resent-From: Daniela Nikolova 
Resent-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.