From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: A reference question
Date: 20 Feb 1999 05:41:02 GMT
Newsgroups: sci.math
Keywords: groups that have a unique normal subgroup
Phd1993jh wrote:
>Can anyone think of any book, article, website, or whatever that discusses
>groups that have a unique *nontrivial* normal subgroup?
I don't know how much detail you want. You're looking for groups G with a
normal subgroup N such that 1 < N < G, and with no normal subgroups besides
these 3, right? Well, that means G/N must be simple, since H/N normal in
G/N implies H normal in G. Also if N had a characteristic subgroup K,
it'd be normal in G, contrary to assumption; so N is characteristically
simple, that is, N is a finite product of simple groups (um, did you mean
for G to be finite?). But now, conversely, if G/N is simple and N
characteristically simple, then G itself has the desired property unless
N splits into a product of nontrivial G-invariant subgroups.
So it seems to me you're just asking for a description of all extensions
1 -> K^n -> G -> Q -> 1 with K and Q (finite?) simple groups, in
which you allow any irreducible action of Q on K^n . Except in the case
K = Z/pZ, elementary cohomology theory shows there is a unique such extension
for every action of Q on K^n.
Some easy examples are direct products K x Q and wreath products K wr Q
(for arbitrary transitive permutation representations of Q).
From memory I think there is something about this in Suzuki's book?
dave