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    finite Abelian group

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    Suppose that G is a finite Abelian group and G has no element of order 2. Show that the mapping g-->g^2 is an automorphism of G. Show, by example, that if G is infinite the mapping need not be an automorphism (hint: consider Z).

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    https://brainmass.com/math/linear-transformation/finite-abelian-group-359983

    Solution Preview

    It is given that G is a finite Abelian group, and has no element of order 2.
    Let us denote the mapping g - > g^2, by f. That is, define f: G -> G as
    f(g) = g^2, for all elements g of G

    In order to show that f is an automorphism, we must show that f is an isomorphism that is onto (that is, we must show that the f is a homomorphism from that is injective, and also surjective - that f(G), the image of G, is G itself).

    Step 1:
    First, prove that f is a homomorphism:
    Let a and b be two elements of G
    Then f(a)f(b) = (a^2)(b^2)
    = (ab)^2 [Since G is Abelian]
    = f(ab)
    Hence, f is a homomorphism.

    Step 2:
    Prove that f is injective:
    Since G has no element of order 2:
    for any element g in G, g^2 = e implies that g = e (where e is the identity element)
    Therefore, f(g) = e implies that g = e [Since f(g) = g^2]
    Thus the kernel of f, is
    ker f = {e} [Definition: ker f = {g in G | f(g) = e}]
    But the kernel of a homomorphism is trivial (consists only of the identity element) if and only if it is injective.
    Hence, f is injective

    Step 3:
    Prove that f is surjective (onto):
    Since G is finite, and f is injective:
    f is an injective function between two sets of the same cardinality (the "two" sets here being G itself)
    Therefore, f is surjective

    Thus, f is an automorphism of G.
    QED.

    Now we have to show, by example, that f need not be an automorphism if G is infinite.
    Consider the set of integers Z, under the operation of addition. It is easily shown that Z forms a ...

    Solution Summary

    The solution shows that the mapping g-->g^2 is an automorphism of G. Show, by example, that if G is infinite the mapping need not be an automorphism (hint: consider Z).

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