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Math Help - Normal subgroups

  1. #1
    Junior Member
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    Bogota D.D
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    Normal subgroups

    Hi everyone.


    Let N be a normal subgroup of G of finite index n. Given an element t\in{G}, let h be the least positive integer such that t^h\in{N}. Prove that h|n; also show that, if t is if finite order r, then h|r

    Thanks.
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  2. #2
    MHF Contributor

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    Tejas
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    Re: Normal subgroups

    consider the mapping G→G/N given by g→gN.

    then t→tN, and so tk→tkN.

    if 0 < k < h, then tkN ≠ N (since tk is not in N).

    but tkN = (tN)k, so tN has order h in G/N.

    so h divides |G/N| = [G:N] = n.

    since g→gN is a homomorphism, the order of gN divides the order of g, if the order of g is finite.

    thus the order of tN divides the order of t, that is h divides r.
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