# Show that do not exist $G$ [duplicate]

Possible Duplicate:
There does not exist a group $G$ such that $|G/Z(G)|=pq$ for $p,q$ prime.

Let $p$ and $q$ prime numbers, with $p<q$ and $p \nmid (q-1)$. Show that do not exist group $G$ where $$\left\lvert\frac{G}{Z(G)}\right\rvert=pq.$$

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## marked as duplicate by Douglas S. Stones, DonAntonio, David Wallace, userNaN, Noah SnyderOct 17 '12 at 10:53

Use this Fact: If $G/Z(G)$ is cyclic, then G is abelian.