Let g be a group and H a subgroup of G.For a fixed a∈G,a coset of H in G is a subset of G of the form aH={ah:h∈H} (Note:aH is only a subset usually and is only a subgroup if a∈H).

(a) Prove that f:H→aH given by f(x)=ax is a bijective function.

(b) Prove that |H|=|aH|for all a∈G.

(c) For any a_1,a_2∈G,prove that either a_1 H=a_2 H or a_1 H∩a_2 H=ϕ.

(d) Let a_1 H,a_2 H,...,a_n H be the distinct cosets of H in G.Prove that G=⋃_(i=1)^n▒〖a_i H〗.

(e)Prove that |G|=n|H|,and hence that |H|| |G|.