1) Do you know the definition of the centralizer of g?

2) Do you know the definition of the center of a group?

3) Do you know the definition of a coset?

These two problems are just straightforward consequences of the definitions.

Maybe this will help: a is in bH (where H is a subgroup of some group G, a and b are elements of G) if and only if (b-inverse)(a) is in H.