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    Group Homomorphism and Abelian Groups

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    Let phi: G ---> H be a group homomorphism. Show that phi[G] is abelian if and only if for all x, y in G, we have xyx^(-1)y^(-1) in ker(phi).

    Proving (=>) seems almost obvious since if it is abelian that means xyx^(-1)y^(-1) = xx^(-1)yy^(-1)=ee which is in the kernel. Please show how to do the reverse (<=) and show that phi is abelian.

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    "=>": If phi(G) is abelian, then ...

    Solution Summary

    Group homomorphism and abelian groups are investigated in the solution.