I need some serious help with these questions. I have no clue where to start:

(G denote a group)

1. Let G be cyclic of order 36 and a be in G with a^12 not equal to, e (e is not equal to both of them), and not equal a^18. Prove G = <a>.

2. Let G be abelian of order p*g, where p and g are different primes. Show G is cyclic.

3. Let sigma be the following permutation in S_6: sigma = (134)(1562). Find sigma^38. (Write as a permutation not a product of cycles.)

4. Suppose k is a proper subgroup of H and H is a proper subgroup of G. If K=|420| and G=|42| what are th possible orders of H? Provide Arguments.

Any help and all hints are appreciated. Thanks guys!!!