Thread: Left and Right Cosets

Let G be a group and H a subgroup of G. Let a, b in G. Prove the following:

If aH = Ha and bH = Hb, then (ab)H = H(ab).

Can someone show this please? Thanks a lot in advance!

Originally Posted by jzellt

Let G be a group and H a subgroup of G. Let a, b in G. Prove the following:

If aH = Ha and bH = Hb, then (ab)H = H(ab).

Can someone show this please? Thanks a lot in advance!

So we want to show that

let

Then for some

By the relation we know that there exists an such that

and we know that by there exists an such that

Putting all of these together we get



Now you just need to show or justify the other direction (it is exactly the same.)