Permutions that commute

Question

So I have two permutations for which i worked out the product disjoint cycles.

f = (1 2 4)(3)(5) and h = (1)(2 4)(3)(5), i also worked out the order of both,

but how can you show that they commute if they do at all.

I know this is basic, Im just a beginner.

score 1 · Accepted Answer · answered Oct 18 '16 at 11:57

1

You don't have to calculate the complete composition but just look at the image of one element, see if it gives a difference and end, or take another point if equal. In your example start with $1$, then $f(1) = 2$ and $h(2) = 4$, on the other hand $h(1) = 1$ and $f(1) = 2$ so $hf(1) = 4$ and $fh(1) = 2$, So $fh \neq hf$.

answered Oct 18 '16 at 11:57

Marc Bogaerts

6,197

First of all you can drop all one-cycles, so $f = (1,2,4)$ and $h = (2,4)$. If you evolve more in the matter you will automatically recognize situations where permutations commute, such as those that work on different sets of points : $f = (1,2,3)$ and $h = (4,5,6)$, or those that are powers of one another : $h = f^2 = (1,3,2)$. – Marc Bogaerts Oct 18 '16 at 13:04

score 0 · Answer 2 · answered Oct 18 '16 at 11:24

0

Just compute $fh$ and $hf$ and see if they are equal.

answered Oct 18 '16 at 11:24

lhf

216,483

Permutions that commute

2 Answers2