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# Isomorphisms, Cyclic Groups and Groups of Permutations

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

We would like to find a group of permutations that is isomorphic to ZZ.
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 ZxZ
has the same order. And then check that their identity elements have the same order?

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https://brainmass.com/math/combinatorics/isomorphisms-cyclic-groups-groups-permutations-107387

#### 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.

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