Order, periods, and cosets

Let be a group of order 36 and let be a normal subgroup of order 6. Let be an element of period 4.

a) Explain why

Because the period of any element must divide the order of the gorup that the element is in. 4 | 36 so but 4 does not divide 6 so

b) What is the order of the factor group

order is 6 because order divides order = 6

c) Work out that is trivial in

, which is the trivial coset

d) Prove has period 2 in

...

e) Prove that

...

For some reason when it comes to doing periods of cosets I get confused. Can anyone help me out?

Thanks in advance,