tonicopm

Dear all:
Let's say G is an h-group if for any natural number n there
exists at most a finite number of subgroups(=sbgps.) of G of index
n (for example finitely generated groups are h-groups). It's easy to
show that G is an h-group iff for any normal sbgp. K of G of finite
index (=f.i.) and for any finite simple group S there exists only a
finite number of normal sbgps. L of K s.t. K/L is isomorphic to S.[1]

So let be given an h-group G, a f.i. normal sbgp. K of G and a
finite simple group S, and let L1,..., Lr be all the normal sbgps.
of K s.t. K/Li isomorphic to S. We can see at once that for all pair
of different indexes 1 <= i,j <= r we have that LiLj = K.
My question is: can we bound somehow, probably by imposing some
additional conditions on G, or on K, the number r in terms of n?

For example, we could demmand that K/Li be a chief factor of G (thus
Li is a normal sbgp. OF G which is maximal with respect to being
contained in K), and if G is say solvable, or even locally solvable,
then S would be abelian...or we could begin by supposing that n is
the least natural number for which there is a normal sbgp. K with
some quotient isomorphic to S; we could even begin at first
considering sbgps. of index AT MOST n, for some natural n, or we
could consider only sbgps. Li which are normal IN G, etc.

If, for example, we take K = G and we ask how many different normal
sbgps. Li of G are there s.t. G/Li is isom. to S, then automatically
G/Li is a chief factor of G and thus we should, probably, ask about
these factors under so and so conditions.
The above question popped up pretty surprisingly during my PhD
research and I didn't pay much attention to it back then. Any
insight, hint, recommendation, quote, etc. will be much appreciated.
Saludos
Tonio

[1] Wilson, John S., "Groups Satisfying the Maximal Condition for
Normal Subgroups", Math. Z. 118, 107-114 (1970), Lemma 1.

mareg

In article <8c09f9d2.0402100117.3ee49780@posting.google.com>,
Tonicopm@yahoo.com (Tonio) writes:

What exactly is n here please?
The only n I can see is in the first papragraph, where it is
quantified by a `for all'.
I cannot tell whether it is |G:K|, |G:Li|, or something else.

Derek Holt.

tonicopm

***********************************
Hi Derek:
First, thanx for even reading my post. Now, n is the index of (the
normal sbgop.) K in G, so actually n = |G/K|. I'm trying to find out
whether there's some kind of relation between this n and r = the
number of normal sbgps. L of K s.t K/L is isomorphic to S, for some
given finite simple group S. I proposed in my 1st. message that n
could be somehow restrcited, e.g. assuming n is the least natural s.t.
has an normal sbgp. K which has some normal sbgp. L s,t, K/L is iso.
to the given S, or some other suitable condition that could be found.
So no a priori conditions whatsoever on n, or even on G (except
trivial cases, like G being finite or even f.g.)...that's all.
The problem arised looking at the lemma quoted in the paper of Wilson.
Be well and thanx
Tonio