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    Abstract Algebra: Three Problems about Homomorphisms, Isomorphisms, and Automorphisms of Groups.

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    Problem 1. Prove that Z / <n> ≈ Z_n , where n ∈ Z and n > 1.

    Problem 2. Prove that θ : g --> a^{-1} ga for a fixed a ∈ G and all g ∈ G defines an automorphism of G.

    Problem 3. Prove if H is the only subgroup of order n in a group G, then H is a normal subgroup of G.

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    Solution Preview

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    PROBLEM 1. Prove that w Zn, where n E Z and n > 1.

    SOLUTION. We define Zn to be the set of integers Zn : {O, 1, 2, - - - ,n — 1} together with the
    addition rule that pirq : 1:, where T is the smallest non-negative integer so that p + q — T is a
    multiple of n. Sincep + q Z O for p, q E {O, 1, 2, - -- ,n -1}, T is just the remainder ofp + q after
    division by n. For example, in Z5, 3+4 : 2. Is is easy to verify that (Zn, Jr) is an abelian group.

    Now, we define the subgroup in Z to be:
    (n) ...

    Solution Summary

    This solution explains the answers for three problems in group theory. The first characterizes the group of integers modulo n in terms of equivalence classes of integers, the second shows that conjugation by a fixed element in a group is an automorphism, and the third uses the result of the second to show that a subgroup is normal if there are no other subgroups of the same order. The solution is available in a .pdf attachment.