Let H and K be subgroups of a finite group G. Let H∩K be a subgroup of G.

(a) For any elements a and b of H, prove that a(H∩K) = b(H∩K) if and only if ak = bK.

(b) Deduce from (a) that the number of elements in the set HK is: |H| |K| / |H∩K|.

(c) If [G:H] and [G:K] have greatest common divisor 1, prove that G = HK and [G:H∩K] = [G:H] [G:K]

Any help would be greatly appreciated!