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