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Math Help - Homomorphism and order of groups

  1. #1
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    Homomorphism and order of groups

    Question: If f: G \rightarrow H is a homomorphism and gcd(\mid G \mid, \mid H \mid ) = 1, prove that f(x) = 1, \forall x \in G.

    I am able to write down some simple results but is unable to piece them together. I have the following:
    1) Since ker(f) is a subgroup of G and im(f) is a subgroup of H, gcd(\mid ker(f) \mid, \mid im(f) \mid ) = 1. This is acheived after some manipulation.

    2) Since f is a homomorphism, order of f(x) divides the order of x

    3) I tried to work towards showing that kef(f) = G as this will result in f(x) = 1, \forall x \in G. But could not get anywhere close to it.

    4) Alternatively, i tried to show that im(f) = {1}. Similarly, I could not get anything.

    So is there anything that i missed out? Thank You.
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  2. #2
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    Re: Homomorphism and order of groups

    f(G) (or im(f) as you call it) is a subgroup of H, so |f(G)| divides |H|.

    let K = ker(f), then |f(G)| = |G|/|K| (since f(G) is isomorphic to G/ker(f), by the FIT).

    thus |K||f(G)| = |G|, so |f(G)| divides |G|. this implies |f(G)| divides gcd(|G|,|H|).
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