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?
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.