Yes, I did mean the direct product.
I also thought that this might be true, but it is not, as shown by the
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; true
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 <firstname.lastname@example.org>
Sent: Monday, October 07, 2002 10:34 AM
Subject: Self-normalized subgroups
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,