26 Dec
2013
26 Dec
'13
2:44 p.m.
On Thu, Dec 26, 2013 at 2:27 PM, Eugene Salamin <gene_salamin@yahoo.com>wrote:
Let G be a group, and let S be the permutation group on G. For a in G, let L(a) be the permutation that sends x to ax, and let R(b) be the permutation that sends x to xb' (' denoting inverse). Then im L and im R are isomorphic copies of G in S, and each is the centralizer in S of the other. I have a vague memory of having seen a proof of the statement concerning centralizers, but can't pin it down. Can someone point me to either a proof, or a counterexample?
If you want a reference you can start here: http://en.wikipedia.org/wiki/Holomorph_%28mathematics%29#Hol.28G.29_as_a_per...