# Why write permutations as disjoint cycles and transpositions?

Apologies in advance if this isn’t an appropriate question for this website. I was reading an introduction to groups book which explained you can write permutations as a product of disjoint cycles and then further that you can write a cycle as a product of transpositions. What is the benefit of thinking about a permutation this way? I do kind of see that by grouping the numbers into the cycles, you can see which numbers are involved with each other and which numbers are unaffected by the rearrangement the permutation is describing.

• as you proceed further in your studies you may be tasked on finding the order of a permutation or determining whether a permutation is in the alternating group. these are questions which can easily be answered when using disjoint cycle notation/decomposing them as a product of transpositions. – thesmallprint Jul 14 '18 at 22:21
• Disjoint cycles just makes the permutation easier to deal with since it is in its 'simplest form'. The decomposition into transpositions is only useful for determining the parity of the permutation (or whether it is odd or even, depending on which definition you use) and this becomes important when you consider groups of permutations. In particular, we have that $S_{n}$ is the set of all permutations on $n$ elements, and that $A_{n}$ is the set of all even permutations on $n$ elements. The group $A_{n}$ is of great importance in classifying some groups of particular orders, for example. – Bill Wallis Jul 14 '18 at 22:24
• It's the same reason we write whole numbers as the product of their prime factors. We don't always do it, but sometimes it helps a lot! – dbx Jul 14 '18 at 22:30
• This is a useful presentation in some contexts. For instance: prove that every permutation is a composition of two involutions. – Michal Adamaszek Jul 14 '18 at 22:44
• I see, thanks, everyone! I didn’t think about cycles like factoring a permutation, but I can see how it is similar. – anonanon444 Jul 14 '18 at 23:44

## 1 Answer

Factoring into disjoint cycles (a transposition is a cycle) is usually the most "useful" way of looking at a permutation. Two main reasons (though there are probably lots of others):

1. It makes it very clear what the orbits of a permutation are.
2. It makes it very easy to compute the order of the permutation (LCM of the cycle lengths).