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 $\displaystyle c \in G $ such that $\displaystyle cH=bH$, then $\displaystyle Hc = Hb$

3. If $\displaystyle f \in G $ such that $\displaystyle Hf=Hb $, then $\displaystyle fH=bH$

My Proof.

1 to 2:

Let x be in cH, then there exist an element u in H such that $\displaystyle x=cu$. Since $\displaystyle cH=bH$, there exist an element v in H such that $\displaystyle x=cu=bv$. Since $\displaystyle bH=Hb$, there exist an element w in H such that $\displaystyle x=cu=bv=wb$.

Now, suppose that y is in Hc, then there exist an element d in H such that $\displaystyle y=dc$.

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

Thanks.