This theorem is taken from Contemporary Abstract Algebra, Ed. 7.
If G is a group and |G|=p^(2) where p is a prime, then G is isomorphic to Z(p^2) or
Z(p) + Z(p).
Note: Here Z(n) is the group under addition modulo n. Also, if G and H are groups, then G + H represents the external direct product of G and H.
Suppose G is not isomorphic to Z(p^2). Then G does not have an element of order p^2. For if G had an element of order p^2, then G is isomorphic to Z(p^2), a contradiction. (I omit some details here since I understand it). Thus by Lagrange Theorem, every non-identity element of G must have order p.
(The next claim, I am not able to understand)
Claim: for any a in G, the subgroup <a> is normal in G
Note: <a> means the group/subgroup generated by a
I am very close to undertanding the book's proof, but there are some missing details that I do not seem to be able grasp.