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    ? Let G be a group and let a,b be two elements of G. The conjugate of b by a is, by definition, the element . The centralizer of a, denoted by s the set of all elements g in G such that ga=ag.

    i) Find all possible conjugates f the permutation

    ii) Find the centralizer p in .
    iii) Prove that for any element a in a group G, the centralizer is a subgroup of G.

    ? Let G= , the non zero real numbers, with operation given by multiplication. Let H be , the positive real numbers, with operation given by multiplication. You may assume that G and H are groups. Let be the map given by .

    i) Prove that is a group homomorphism.
    ii) Compute ker and im , explaining your answer

    *Please see attachment for complete list of questions

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    Solution Summary

    This shows hwo to find all possible conjugates, find the centralizer, compute kernel and image, and prove centralizer is a subgroup.