Element structure of alternating group:A6

This article gives the element structure of alternating group:A6.

See also element structure of alternating groups and element structure of symmetric group:S5.

For convenience, we take the set acted upon as $$\{ 1,2,3,4,5,6 \}$$.

Order computation
The alternating group of degree six has order 360, with prime factorization $$360 = 2^3 \cdot 3^2 \cdot 5^1 = 8 \cdot 9 \cdot 5$$. Below are listed various methods that can be used to compute the order, all of which should give the answer 360:

Conjugacy class structure
There is a total of 7 conjugacy classes, of which 5 are unsplit from symmetric group:S6, and 2 are a split pair arising from a single conjugacy class in $$S_6$$. The conjugacy class sizes are 1, 40, 45, 90, 40, 72, 72.

Interpretation as alternating group
For a symmetric group, cycle type determines conjugacy class. The statement is almost true for the alternating group, except for the fact that some conjugacy classes of even permutations in the symmetric group split into two in the alternating group, as per the splitting criterion for conjugacy classes in the alternating group, which says that a conjugacy class of even permutations splits in the alternating group if and only if it is the product of odd cycles of distinct length.

Here are the unsplit conjugacy classes:

Here is the split pair of conjugacy classes:

Interpretation as projective special linear group of degree two
Compare with element structure of projective special linear group of degree two over a finite field.

We consider the group as $$PSL(2,q)$$, $$q = 9$$. We use the letter $$q$$ to denote the generic case of $$q \equiv 1 \pmod 4$$.

Number of conjugacy classes
The alternating group of degree six 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: