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    Isomorphisms, Cyclic Groups and Groups of Permutations

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    Consider the group Z[4] × Z[6] under * such that
    (a, b) * (c, d) = (a +[4] c, b +[6] d).
    (here +[4] means + is in Z[4] and +[6] is in Z[6])

    We would like to find a group of permutations that is isomorphic to Z[4]Z[6].
    Is this group cyclic? If so, prove it. If not, explain why.

    Do I need to list all the members and check or is it enough to know that Z[2]xZ[12]
    has the same order. And then check that their identity elements have the same order?

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    Solution Preview

    Let a=(1 2 3 4) and b=(5 6 7 8 9 10) be two cycles in the permutation
    group S10. a and b are two disjoint cycles. Let G=<a,b> be the subgroup
    generated by a and b.
    Now I show that G is isomorphic to Z4xZ6.
    Since a and b are disjoint, then ab=ba. So each ...

    Solution Summary

    Isomorphisms, Cyclic Groups and Groups of Permutations are investigated. The solution is detailed and well presented.