Centralizers in topological groups
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From: bell.charles62 on 3 Aug 2010 16:02 Dear members, Let G be a topological (Hausdorff) group. Let H be an "abstract" subgroup of G (i.e., not neceessary closed, open, ...) and let K denote its topological closure in G. Is it true that C_G(H) is contained in C_G(K)? What about normalizers? I'm sorry if these are too stupid questions. Best wishes, Charles.
From: Arturo Magidin on 3 Aug 2010 16:33 On Aug 3, 3:02 pm, "bell.charle...(a)googlemail.com" <bell.charle...(a)googlemail.com> wrote: > Dear members, > > Let G be a topological (Hausdorff) group. Let H be an "abstract" > subgroup of G (i.e., not neceessary closed, open, ...) and let K > denote its topological closure in G. > Is it true that C_G(H) is contained in C_G(K)? What about normalizers? > I'm sorry if these are too stupid questions. Conjugation by an element g is a topological automorphism of G; in particular, if n lies in the normalizer of H, then conjugation by n maps H to itself, and thus maps any closed subset that contains H into a closed subset that contains H. In particular, nKn^{-1} must contain G and is closed, and therefore K<=nKn^{-1}. The same argument holds for n^{-1}, yielding that K<=n^{-1}Kn, and therefore nKn^{-1} = K, showing that n normalizes K as well. So N_G(H)<=N_G(K). For the centralizer, note that for any k in K there is a sequence {g_i} of elements of G that converges to k; again apply conjugation by a centralizing element, and recall that in a Hausdorff space sequences that converge will converge to a unique point. -- Arturo Magidin
From: bell.charles62 on 3 Aug 2010 18:57
Dear Mr. Arturo, Thank you very much for your reply. I realized that my question on centralizers follows from the following fact: "two functions coinciding on a subset must also coincide in the closure of that set". So nets (or sequences) are not really needed but the Hausdorff property is fundamental. Best wishes, Charles. On Aug 3, 5:33 pm, Arturo Magidin <magi...(a)member.ams.org> wrote: > On Aug 3, 3:02 pm, "bell.charle...(a)googlemail.com" > > <bell.charle...(a)googlemail.com> wrote: > > Dear members, > > > Let G be a topological (Hausdorff) group. Let H be an "abstract" > > subgroup of G (i.e., not neceessary closed, open, ...) and let K > > denote its topological closure in G. > > Is it true that C_G(H) is contained in C_G(K)? What about normalizers? > > I'm sorry if these are too stupid questions. > > Conjugation by an element g is a topological automorphism of G; in > particular, if n lies in the normalizer of H, then conjugation by n > maps H to itself, and thus maps any closed subset that contains H into > a closed subset that contains H. In particular, nKn^{-1} must contain > G and is closed, and therefore K<=nKn^{-1}. The same argument holds > for n^{-1}, yielding that K<=n^{-1}Kn, and therefore nKn^{-1} = K, > showing that n normalizes K as well. So N_G(H)<=N_G(K). > > For the centralizer, note that for any k in K there is a sequence > {g_i} of elements of G that converges to k; again apply conjugation by > a centralizing element, and recall that in a Hausdorff space sequences > that converge will converge to a unique point. > > -- > Arturo Magidin |