Prove the Followings
1) Let H, K, HK?G. Then |H:H?K| ? |G:K|.
If |G:K| < ? then |H:H?K| = |G:K| <=> G = HK ?
2) Let H, K be subgroup of finite index of G and HK ? G
Then |G:H?K| is finite and |G:H?K| ? |G:H||G:K|
Furthermore |G:H?K| = |G:H||G:K| <=> G = HK ?
3) Let H, K, HK ? G, |G:H| = m, |G:K| = n and
(m, n) = 1 then G = HK ?
Recall: Let H, K, HK ? G
a) HK = KH (Since HK ? G)
|HK:H| = |KH:H| = |K:H?K|
|KH:K| = |HK:K| = |H:H?K|