Problem involving centralizer of the group?

Suppose A belongs to a group and |A|=5. Prove that C(A) = C(A^3). Find an element a from some group such that |A|=6 and C(A) doesnt equal C(A^3) I understand C(A) = { gεG : gA = Ag}.....so g and A commute, but I havent seen a good example of a problem like this I know A^5 = e...so does saying C(A) = C(A^3)... show more Suppose A belongs to a group and |A|=5. Prove that C(A) = C(A^3). Find an element a from some group such that |A|=6 and C(A) doesnt equal C(A^3)

I understand C(A) = { gεG : gA = Ag}.....so g and A commute, but I havent seen a good example of a problem like this

I know A^5 = e...so does saying C(A) = C(A^3) imply that a^15=e? I would guess so...or at least something along those lines. also a general question....if set/group has a finite integer order, does that mean it is cyclic?
Thanks a lot
Update: How does that relate to part b?
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