Write (13257)(23)(47512) as a product of disjoint cycles

[Mr Davis 97]

Homework Statement

Write (13257)(23)(47512) as a product of disjoint cycles. Each bracket is a permutation of seven elements written in cycle notation.

Homework Equations

The Attempt at a Solution

This isn't too hard of a problem. One easy way would be to evaluate the entire product, and then write that product in cycle notation. However, is there an easier, faster way of doing this, just be looking at the expression (13257)(23)(47512) directly?

 

[fresh_42]

Not that I know of. And $2$ occurs in all three cycles, so any possible "rule" is likely more complicated than simply multiply them.

 

[Mr Davis 97]

Also, one quick related question. I need to calculate the order of (125)(34). Is there a quick way to do this, or do I have to literally keep composing the permutation with itself until I get the identity permutation?

 

[fresh_42]

If they are disjoint, they commute. So $(ab)^n=a^nb^n$ and the order is the least common multiple of both orders. And a cycle of length $n$ is of order $n$.