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    Subgoups : Indicies

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    Note: C means set containment (not proper set containment), |G : K| means index of subgroup K in G, and G # K means K is a normal subgroup of G

    question:
    Let K C H C G be groups, where K # G and |G : K| is finite. Show that |G/K : H/K| is also finite and that |G/K : H/K|=|G : H|

    © BrainMass Inc. brainmass.com March 4, 2021, 5:53 pm ad1c9bdddf
    https://brainmass.com/math/linear-algebra/subgoups-indicies-15974

    Solution Preview

    Proof:
    Since KCHCG and |G:K| is finite, then |G:H| is finite. K is a normal subgroup of G, then H/K is a subgroup of G/K. Suppose |G:H|=n, then we have
    G=g1H U g2H U...U gnH, where U ...

    Solution Summary

    A proof involving subgroups and indicies is offered. The proof is concise.

    $2.19

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