# cosets

• November 6th 2007, 12:45 PM
chris27
cosets
Could someone help me with this proof? Thanks!

Let G be a group, and H a subgroup of G.
Ha = Hb iff ab^-1 is in H
• November 6th 2007, 01:27 PM
Soltras
Quote:

Originally Posted by chris27
Could someone help me with this proof? Thanks!

Let G be a group, and H a subgroup of G.
Ha = Hb iff ab^-1 is in H

$Ha = Hb \Leftrightarrow Hab^{-1} = Hbb^{-1} \Leftrightarrow Hab^{-1} = He \Leftrightarrow Hab^{-1} = H \Leftrightarrow ab^{-1} \in H$.
• November 6th 2007, 01:59 PM
ThePerfectHacker
I do not like your proof Soltras.

$Ha = Hb$ now $a\in Ha$ so $a\in Hb$ that means $a=hb$ for some $ab^{-1} = h \in H$. The converse if almost identity.