> < ^ From:

> < ^ Subject:

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; 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

self-normalized.

-Avital.

----- 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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