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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)

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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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