Element structure of symmetric group:S5

This article gives information about the element structure of symmetric group:S5.

See also element structure of symmetric groups.

Family contexts
Note that if you go to the section of this article, you'll find a discussion of the conjugacy class structure with each of the below family interpretations.

Order computation
The symmetric group of degree five has order 120, with prime factorization $$120 = 2^3 \cdot 3^1 \cdot 5^1= 8 \cdot 3 \cdot 5$$. Below are listed various methods that can be used to compute the order, all of which should give the answer 120:

Interpretation as symmetric group
For any symmetric group, cycle type determines conjugacy class, i.e., the cycle type of a permutation (which describes the sizes of the cycles in a cycle decomposition of that permutation), determines its conjugacy class. In other words, two permutations are conjugate if and only if they have the same number of cycles of each size.

The cycle types (and hence the conjugacy classes) are parametrized by partitions of the size of the set. We describe the situation for this group:





Here is some more information:

Interpretation as projective general linear group of degree two
Compare with element structure of projective general linear group of degree two over a finite field. Here, the field is field:F5, so $$q = 5$$.

Number of conjugacy classes
The symmetric group of degree five has 7 conjugacy classes. Below are listed various methods that can be used to compute the number of conjugacy classes, all of which should give the answer 7: