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Isomorphisms, Cyclic Groups and Groups of Permutations

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?

Solution Preview

Proof:
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. The response received a rating of "5/5" from the student who originally posted the question.

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