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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, Norbert, Noah Snyder Oct 17 '12 at 10:53

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Isn't this the same as… which you asked yesterday? – Gerry Myerson Oct 17 '12 at 1:00

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

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You asked the very same question and you also wrote the very same comment. The answer, as it is, is enough for you to develop a solution. If there's something you don't understand then ask. – DonAntonio Oct 17 '12 at 3:29
@JarbasDantasSilva What's the center of an abelian group? – Alexander Gruber Oct 17 '12 at 4:04
the group...???????????? – Jarbas Dantas Silva Oct 17 '12 at 16:14

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