# Math Help - cosets

1. ## 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

2. 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
$Ha = Hb \Leftrightarrow Hab^{-1} = Hbb^{-1} \Leftrightarrow Hab^{-1} = He \Leftrightarrow Hab^{-1} = H \Leftrightarrow ab^{-1} \in H$.

3. 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.