# Suppose that $G$ is non-abelian group and $|G|=pq$ such that $p$ and $q$ are prime numbers [closed]

Suppose that $$G$$ is a non-abelian group and $$|G|=pq$$ such that $$p$$ and $$q$$ are prime numbers and $$N$$ is normal subgroup of $$G$$ such that $$|N|=q$$ . show that $$G'=N$$.

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## 1 Answer

$$|G/N|=p$$ so $$G/N$$ is cyclic $$\Rightarrow1\not= G'\subset N\Rightarrow G'=N$$