Exercise 4.3.4 Let p≥ 2 be a prime and prove that any group of order p² is isomorphic to either Zp² or Zpx Zp.

Hint: If there exists an element of order p², then GZp², so assume that this is not the case and deduce that

every nonidentity element of G has order p. Then prove that for any two elements g, h E G, either (g) = (h) or

(g) n(h) = {e}. Finally, prove that there are two nonidentity elements g, h E G so that (g) n(h) = {e}, and apply

Proposition 3.7.1.