# Linked Questions

41 questions linked to/from Normal subgroup of prime index
4answers
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### $G$ is a finite group such that $|G| = n$ and $p$ is minimal prime dividing $n$. $H \subset G, [G:H] = p$. Prove $H$ is normal in $G$ [duplicate]

Let $G$ be a finite group of order $n$ and $p$ be the minimal prime number dividing $n$. Assume that $H \subset G$ is a subgroup of index $p$. Prove that $H$ is normal: $H \trianglelefteq G$ To do ...
2answers
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### If a subgroup has smallest prime index, then it is normal [duplicate]

Assume that $G$ is finite with $p$ the smallest prime dividing its order. Suppose $H < G$ with $[G:H]=p$. Prove that $H \lhd G$. I've seen this question a few times on here but all the proofs I ...
1answer
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3answers
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### Prove that every group of order 15 is abelian? [duplicate]

I had seen this proof at many places, but everywhere sylows theorem is used. So is their any way to solve it without using sylows theorem?

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