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    Groups and proofs

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    Let A, B, and C be groups, let alpha, beta, and gamma be homomorphisms with gamma times alpha = beta

    alpha gamma beta

    If alpha is surjective, prove that ker(gamma)= alpha((ker(beta)).

    Prove that if K is a subgroup of a group G, and if every left coset of aK is equal to the right coset Kb, then K is a normal subgroup of G.

    I just dont know how to do these, the confuse me and I am horrible at writing them in good proof form

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    Solution Preview

    Please see the attachment.

    Problem #1
    From the condition, we have the following relations.
    We have .
    Now suppose is surjective, we want to show that
    For any , we have , then . ...

    Solution Summary

    This contains proofs regarding a surjective homomorphism and a normal subgroup.