Question
Suppose that $N$ is a normal subgroup of a finite group $G$ and $H$ is a subgroup of $G$. If $|G / N|$ is prime, prove that $H$ is contained in $N$ or that $N H=G$.
Step 1
Since $N$ is normal in $G$, we have that $N \trianglelefteq NH$. Now, consider the quotient group $(NH)/N$. By the Second Isomorphism Theorem, we have that $(NH)/N \cong H/(H \cap N)$. Show more…
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