Problems:
Normal subgroups, Second Theorem of Isomorphism, Conjugates and Cyclic Groups
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).