> < ^ Date: Mon, 07 Oct 2002 02:57:16 -0500
> < ^ From: Avital Oliver <olivera@macs.biu.ac.il >
> < ^ Subject: Re: Self-normalized subgroups

Yes, I did mean the direct product.

I also thought that this might be true, but it is not, as shown by the
following example:

gap> G := DirectProduct(SymmetricGroup(3), SymmetricGroup(3));
Group([ (1,2,3), (1,2), (4,5,6), (4,5) ])
gap> H := Group([(1,2,3)(4,5,6), (1,2)(4,5)]);
Group([ (1,2,3)(4,5,6), (1,2)(4,5) ])
gap> Normalizer(G, H);
Group([ (2,3)(5,6), (1,2,3)(4,5,6) ])
gap> Normalizer(G, H) = H;

Thus, H is self-normalized in S_3 x S_3, but obviously H is *not* of form A
x B, and even stronger than that, the intersection of H with both of the
S_3's in the direct product is the trivial subgroup, which is not


----- Original Message -----
From: Marco Costantini <costanti@giove.mat.uniroma1.it>
To: <olivera@macs.biu.ac.il>
Sent: Monday, October 07, 2002 10:34 AM
Subject: Self-normalized subgroups

Dear Avital,
Does (G x H) means the direct product of G and H? If yes, then I would say
that the subgroups of (G x H) of the form (M x N), where M is a
self-normalized subgroups of G and N is a self-normalized subgroups of H,
are self-normalized subgroups of (G x H). Perhaps all the self-normalized
subgroups of (G x H) are of the above form.
I don't have any better idea about it.
All the best,
Marco Costantini

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