mathematical proof?

Question

show that if p is a prime number, then the abelian groups C_(p^2) and (C_p)x(C_p) are not isomorphic. (even though they have the same number of elements.)

Answer 1

In the second group, any element raised to the pth power equals the identity.

In the first group, that's obviously not true. (In particular, it's not true if the element is a generator for the group.)

Q.E.D.

Or to put it another way, in the first group there's an element of order p^2 by definition, while in the second group the order of every element can quickly be shown to be less than or equal to p.