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    Normal subgroups, Second Theorem of Isomorphism, Conjugates and Cyclic Groups

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    Problem 1.
    Let a,b be elements of a group G
    Show a) the conjugate of the product of a and b is the product of the conjugate of a and the conjugate of b

    b) show that the conjugate of a^-1 is the inverse of the conjugate of a

    c)let N=(S) for some subset S of G. Prove that the N is a normal subgroup of G if
    gSg^-1<=N for all g in G

    d)Show that if N is cyclic, then N is normal in G if and only if for each g in G
    gxg^-1=x^k for some integer k.

    e)let n be positive integer. Prove that the subgroup N generated by all the elements of G of order n is a normal subgroup

    Problem 2.
    Let M and N be normal subgroups of G such that G=MN. Prove that G/(M intersection N) is isomorphic to (G/M)x(G/N).

    (i know that the second problem must be some application of 2nd theorem of isomorphisms but I don't know how to go about it....help)

    © BrainMass Inc. brainmass.com October 9, 2019, 6:44 pm ad1c9bdddf
    https://brainmass.com/math/linear-transformation/normal-subgroups-second-theorem-of-isomorphism-conjugates-and-cyclic-groups-96577

    Solution Summary

    Normal subgroups, Second Theorem of Isomorphism, Conjugates and Cyclic Groups are investigated. The solution is detailed and well presented.

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