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Math Help - normal subgroups and cayley's theorem

  1. #1
    Senior Member abhishekkgp's Avatar
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    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.
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  2. #2
    Senior Member roninpro's Avatar
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    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.

    Try working with this.
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