so I think i got it, but I am not sure!
Let x be a member of G
b=x and a=x
ab^-1=xx^-1 which is a member of G, so e is a member of G
If a=e then b is a member of G
eb^-1 is a member of G so b^-1 is a member of H
x,y are members of G
x=a and y=b^-1
b=y^-1
b is a member of G so ab^-1=xy which is a member of G
Will this work or am i missing some steps?