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Marco Constantini wrote:

Let G be H x K. Then the self-normalized subgroups of G should be

the subgroups S such that the projections of S onto H and K are

self-normalized in H and K respectively.

This condition is neither necessary nor sufficient.

Let H be a subgroup of K. Consider the subgroup S ={(h,h): h in H} of

H X K. An easy exercise shows that S is self-normalized iff the

centralizer of H in K is {e}.

If H = K is a non-trivial abelian group, the both projections are

self-normalized, because they are the entire group K, but S is not

self-normalized.

Conversely, if H= Alt(4) and K = Sym(4), then S is self-normalized,

even though A4 is not self-normalized in Sym(4).

Sincerely,

Luc Teirlinck.

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