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

Neither (why?) nor (why?These two claims are, in

fact, one and the same thing...), but still , thus...
e) Prove that

This follows at once from (c) and from

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

Thanks in advance,