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Given that G is an abelian group and $\Psi: G\to G$ is a homomorphism, what can be said about the kernel of $\Psi$ if $G$ has odd order?

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    I don't think there can be said anything interesting in this generality, except that the kernel has also odd order :) On the other hand finite abelian groups are classified as sums of various $Z/p^n Z$ so one can explicitly write down a list of all possible kernels.2011-12-02

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