From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: Group Theory question
Date: 23 Dec 1999 06:17:22 GMT
Newsgroups: sci.math
Keywords: characteristic subgroups, distinct isomormophic normal subgroups
Martin Green wrote:
>I think if a group has two subgroups that are isomorphic to
>each other, then those subgroups cannot be normal. A normal
>subgroup is therefore one which has no isomorphic images of
>itself within the parent group.
>
>Is this right?
In article <19991222204430.23334.00000541@ng-bg1.aol.com>,
Seraph-sama wrote:
>Almost. If a subgroup H of a group G is "characteristic," then there is no
>subgroup of G other than H that is isomorphic to H. A subgroup of G is
>"characteristic" if it is fixed under every element of Aut G, and in that case
>we say it's Aut G-invariant.
Um, well, we usually say it's ... characteristic!
The first assertion of the last post is incorrect; for example, the dihedral
group G of order 8 has several cyclic subgroups of order 2, one of
which is actually characteristic (the commutator subgroup, G' ). All
automorphisms of G leave G' setwise invariant, true. But there are
isomorphisms between G' and various other subsgroups of G which do
not extend to homomorphisms defined on all of G.
dave