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Let $H$ be a subgroup of $G$ with finite index. Prove that $H$ is characteristic in $G$
Let $G$ be a group and $H$ be a subgroup of $G$ with finite index. Prove that $H$ is characteristic in $G$
My attempt:
By the First Isomorphism Theorem, $G/H$ is isomorphic to $H$. In general, $H$ is normal, so $G/H$ is abelian. The finite index of $H$ is equivalent to the fact that there is some $kinmathbb{Z_+}$ such that $|G|=k|H|$.
By Lagrange’s theorem, $k|H|inmathbb{N}$.
My questions:
What are the other proofs?
What’s wrong with my proof?
A:
Your proof looks fine. It basically shows that the quotient $G/H$ is a finite $|H|$-by-$|H|$ abelian group. Such a group is in particular torsion and in this case, abelian implies it is trivial, i.e. equal to the group.
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