What started me on this path is the easy seen theorem that if G/Z is cyclic, then G is abelian. But then Z = G and G/Z is trivial. But note that G/Z is isomorphic to Inn(G) the inner isomorphism group of G. So no inner isomorphism group can be nontrivial cyclic, a nice fact. -- Gene
________________________________ From: Eugene Salamin <gene_salamin@yahoo.com> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, November 29, 2013 9:13 AM Subject: Re: [math-fun] Group theory question
Thanks people. Confronted with this example, I was able to locate the mistake in my attempt at a proof .
-- Gene
________________________________ From: Cris Moore <moore@santafe.edu> To: wclark@mail.usf.edu; math-fun <math-fun@mailman.xmission.com> Cc: Eugene Salamin <gene_salamin@yahoo.com> Sent: Thursday, November 28, 2013 9:11 PM Subject: Re: [math-fun] Group theory question
Yes, indeed the quaternions are the Heisenberg group where p=2 :-)
On Nov 28, 2013, at 10:09 PM, W. Edwin Clark <wclark@mail.usf.edu> wrote:
The smallest examples are the quaternion group Q8 and the dihedral group
D_8 of order 8.
See http://en.wikipedia.org/wiki/Quaternion_group. The center of Q8 is {1,-1} and the quotient has order 4 so must be abelian.
On Thu, Nov 28, 2013 at 11:48 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
Is the following a theorem?
Let G be a group with center Z. If G/Z is abelian, then G is abelian.
-- Gene