If every right coset of H in G is a left coset of H in G prove that

aHa^-1 = H for all a elements of G?

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- Sep 22nd 2009, 01:23 PMGodisgoodCosets
If every right coset of H in G is a left coset of H in G prove that

aHa^-1 = H for all a elements of G? - Sep 22nd 2009, 02:23 PMHallsofIvy
- Sep 22nd 2009, 02:40 PMTaluivren
Hi,

"every right coset of H in G is a left coset of H in G" means that there exists some b in G such that aH=Hb. so aHb^-1 = H and because the identity element e belongs to H we have ab^-1 in H. Since H is a group, we also have (ab^-1)^-1 = ba^-1 in H. In other words, Hba^-1 = H.

Now we have aHa^-1 = Hba^-1 = H.