Purchase Solution

Isomorphisms, Cyclic Groups and Groups of Permutations

Not what you're looking for?

Ask Custom Question

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?

Purchase this Solution

Solution Summary

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

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 ...

Purchase this Solution

Free BrainMass Quizzes
Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Probability Quiz

Some questions on probability