Let H be a subgroup of G, and b a fixed element of G.

Prove the following statements are equviaent.

1. bH = Hb

2. If such that , then

3. If such that , then

My Proof.

1 to 2:

Let x be in cH, then there exist an element u in H such that . Since , there exist an element v in H such that . Since , there exist an element w in H such that .

Now, suppose that y is in Hc, then there exist an element d in H such that .

I want to get y to look like Hb, but can't really get it...

Thanks.