Why write permutations as disjoint cycles and transpositions?

Question

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.

Answer 1

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).

Answer 2

Shortly

Transpositions are related to the existence of the Alternating Group, within all permutations are written as a product of an even number of transpositions. For example, (1,2,3) = (2,3)(1,2) is even.

The Alternating Group order 60 containing only even permutations, seen as Cayley group, is the smallest simple group, not having normal subgroups.

The above means further that the general equation of degree 5 can not be solved by radicals.