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    Cyclic group

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    Let X be a prime. Prove or disprove that is cyclic for each normal subgroup K.

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    Solution Preview

    First, we should know the definition of Dp.
    Dp=<a,b> is generated by two elements a and b, where
    a^2=b^p=1, ab=b^(-1)a.
    So K=<b> is a normal subgroup of Dp. Because aba^(-1)=b^(-1) is in K.
    Now I claim that K is the unique normal subgroup of Dp if p>=3 is a prime.
    We know, |Dp|=2p, |K|=p, so [Dp:K]=2 and Dp = K union aK. So each ...

    Solution Summary

    This is a proof regarding a cyclic group.