A finite group of width two has a trivial center

Let G be a group. Prove that if |G| = pq for some primes p and q, then either G is abelian (i.e. Z(G) = G) or Z(G) = 1.

Suppose Z(G) \neq 1. Since Z(G) \leq G, by Lagrange’s Theorem, |Z(G)| divides |G|. Thus |Z(G)| is one of p, q, or pq. If |Z(G)| = pq, then since G is finite, G = Z(G) is abelian. If (without loss of generality) |Z(G)| = p, then by Lagrange’s Theorem |G/Z(G)| = q, so that G/Z(G) is cyclic. By a previous theorem, then, G is abelian.

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