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Math Help - order

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    order

    If f: G --> H is an isomorphism, prove that |f(x)| = |x| for all x in G. Deduce that any two isomorphic groups have the same number of elements of order n for each positive integer n. Is it true "if f is a homomorphism, then |f(x)| = |x|?" If true, prove it. If false, give an example.
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    Quote Originally Posted by dori1123 View Post
    If f: G --> H is an isomorphism, prove that |f(x)| = |x| for all x in G. Deduce that any two isomorphic groups have the same number of elements of order n for each positive integer n. Is it true "if f is a homomorphism, then |f(x)| = |x|?" If true, prove it. If false, give an example.
    Hint: Prove that if f: G_1 \to G_2 is an isomorphism then |x| divides |f(x)|. Therefore if f: G_1 \to G_2 is an isomorphism then f: G_1 \to G_2 and f^{-1} : G_2 \to G_1 are homomorphisms therefore |x| divides |f(x)| and |f(x)| divides |x|, so |x|=|f(x)|.
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