Element structure of alternating group:A4

This article gives information on the element structure of alternating group:A4.

See also element structure of alternating groups and element structure of symmetric group:S4.

Summary
The multiplication table (to be completed) is:

Order computation
The alternating group of degree four has order 12, with prime factorization $$12 = 2^2 \cdot 3^1 = 4 \cdot 3$$. Below are listed various methods that can be used to compute the order, all of which should give the answer 12:

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 its cycle decomposition comprises odd cycles of distinct length.

Here are the unsplit conjugacy classes:

In this case, the union of the unsplit conjugacy classes is a proper normal subgroup isomorphic to the Klein four-group. Note that this phenomenon is unique to the case $$n = 4$$.

Here is the split conjugacy class:



Interpretation as projective special linear group of degree two
We consider the group as $$PSL(2,q)$$, $$q = 3$$. We use the letter $$q$$ to denote the generic case of $$q \equiv 3 \pmod 4$$.

Interpretation as general affine group of degree one
The alternating group of degree four is isomorphic to the general affine group of degree one over field:F4. All the elements of this group are of the form:

$$x \mapsto ax + v, a \in \mathbb{F}_q^\ast, v \in \mathbb{F}_q$$

where $$q = 4$$. Below, we interpret the conjugacy classes of the group in these terms:

Number of conjugacy classes
The alternating group of degree four has 4 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 4: