# normal subgroups and cayley's theorem

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• Feb 18th 2011, 08:34 PM
abhishekkgp
normal subgroups and cayley's theorem
Given H is a subgroup of G and that H has finite index n, prove that there is a normal subgroup K of G such that $K \leq H$ and $|G/K| \leq n!$. Don't assume that G is finite.
• Feb 18th 2011, 08:46 PM
roninpro
Note that $G$ acts on the cosets of $H$ by left multiplication. Since there are $n$ cosets, this induces a homomorphism $\phi: G\to S_n$. Then we know that $K=\ker \phi$ is a normal subgroup of $G$.

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