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From: William Elliot on 23 Jul 2010 04:22 Let G be a compact group and U an open nhood of e. Is there an open V nhood e with V subset U and for all a, aVa^-1 = V? A counter example or prove would be nice.
From: A N Niel on 23 Jul 2010 06:11 In article <20100723011837.B85336(a)agora.rdrop.com>, William Elliot <marsh(a)rdrop.remove.com> wrote: > Let G be a compact group and U an open nhood of e. > Is there an open V nhood e with V subset U and > for all a, aVa^-1 = V? > A counter example or prove would be nice. Of course compactness of G is the key.
From: William Elliot on 23 Jul 2010 06:35 On Fri, 23 Jul 2010, A N Niel wrote: > <marsh(a)rdrop.remove.com> wrote: > >> Let G be a compact group and U an open nhood of e. >> Is there an open V nhood e with V subset U and >> for all a, aVa^-1 = V? >> A counter example or prove would be nice. > > Of course compactness of G is the key. > I don't see how to use it.
From: hagman on 23 Jul 2010 15:43 On 23 Jul., 12:35, William Elliot <ma...(a)rdrop.remove.com> wrote: > On Fri, 23 Jul 2010, A N Niel wrote: > > <ma...(a)rdrop.remove.com> wrote: > > >> Let G be a compact group and U an open nhood of e. > >> Is there an open V nhood e with V subset U and > >> for all a, aVa^-1 = V? > >> A counter example or prove would be nice. > > > Of course compactness of G is the key. > > I don't see how to use it. The obvious candidate for V is the intersection of all aUa^-1, a in G. But for it to be open we better intersect only finitely many open sets The aU cover G, hence there is a finite subcover a_1 U, ..., a_n U ... hagman
From: William Elliot on 24 Jul 2010 01:16
On Fri, 23 Jul 2010, hagman wrote: > On 23 Jul., 12:35, William Elliot >> >>>> Let G be a compact group and U an open nhood of e. >>>> Is there an open V nhood e with V subset U and >>>> . . for all a, aVa^-1 = V? > > The obvious candidate for V is the intersection of all aUa^-1, a in G. > But for it to be open we better intersect only finitely many open sets > The aU cover G, hence there is a finite subcover a_1 U, ..., a_n U ... You're suggesting take .. . V = a1.Ua1^-1 /\../\ a_n.U.a_n^-1 ? . . (1) Yes, V is an open nhood of e. Why is V a subset of U? Do we use, .. . for all a, some open U_a nhood e with a.U_a.a^-1 subset U ? Then, by taking .. . V = a1.U_a1.a1^-1 /\../\ a_n.U_an.a_n, . . (2) the desired .. . V open nhood e, V subset U. Yet, why for all a, is aVa^-1 a subset of V? |