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

1. Feb 15, 2017

### Mr Davis 97

1. The problem statement, all variables and given/known data
Write (13257)(23)(47512) as a product of disjoint cycles. Each bracket is a permutation of seven elements written in cycle notation.

2. Relevant equations

3. 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?

2. Feb 15, 2017

### Staff: Mentor

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

3. Feb 15, 2017

### 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?

4. Feb 15, 2017

### Staff: Mentor

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 $n$.