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    Group Theory : Conjugacy, Cayley Table, Subgroups and Quotient Groups

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    Define the notion of conjugacy as it applies in a general group.
    Prove that the inverses of a pair of conjugate elements are also conjugate.
    Prove that conjugate elements have the same order. (6 marks)

    The remainder of this question concerns the group G , whose Cayley table is as

    (b) Determine the inverse and the order of each of the elements of G.

    (c) Simplify each of the following...

    (d) Given that the only element conjugate to g is g itself (you need not prove this), determine the conjugacy classes of G.

    (e) Find H , a normal subgroup of G having three elements. Identify the elements of the quotient group G/H , and determine its isomorphism type.

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    Solution Summary

    Conjugacy, cayley table, subgroups and quotient groups are investigated and discussed. The solution is detailed and well presented.