Theorem 9.7 Classification of Groups or order p^(2)

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.