# Element structure of alternating group:A4

View element structure of particular groups | View other specific information about alternating group:A4

## Summary

Item Value
order of the whole group (total number of elements) 12 (see order computation for more)
conjugacy class sizes 1,3,4,4
maximum: 4, number: 4, sum (equals order of whole group): 12, lcm: 12
See conjugacy class structure for more.
number of conjugacy classes 4
See number of conjugacy classes for more.
order statistics 1 of order 1, 3 of order 2, 8 of order 3
maximum: 3, lcm (exponent of the whole group): 6

The multiplication table (to be completed) is:

## Family contexts

Family name Parameter values General discussion of element structure of family
alternating group degree $n = 4$ element structure of alternating groups
projective special linear group of degree two over a finite field field:F3, i.e., the group is $PSL(2,3)$ element structure of projective special linear group of degree two over a finite field
general affine group of degree one over a finite field field:F4, i.e., the group is $GA(1,4)$ element structure of general affine group of degree one over a finite field
von Dyck group parameters (3,3,2) element structure of von Dyck groups
COMPARE AND CONTRAST: View element structure of groups of order 12 to compare and contrast the element structure with other groups of order 12.

## Elements

### Multiple ways of describing permutations

Cycle decomposition notation One-line notation, i.e., image of string $1,2,3,4$ Matrix (left action)
$()$ 1234 $\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\\end{pmatrix}$
$(2,3,4)$ 1342 $\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\\end{pmatrix}$
$(2,4,3)$ 1423 $\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\\end{pmatrix}$
$(1,2)(3,4)$ 2143 $\begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\\end{pmatrix}$
$(1,2,3)$ 2314 $\begin{pmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\\end{pmatrix}$
$(1,2,4)$ 2431 $\begin{pmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\\end{pmatrix}$
$(1,3,2)$ 3124 $\begin{pmatrix}0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\\end{pmatrix}$
$(1,3,4)$ 3241 $\begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\\end{pmatrix}$
$(1,3)(2,4)$ 3412 $\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\\end{pmatrix}$
$(1,4,2)$ 4132 $\begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\\end{pmatrix}$
$(1,4,3)$ 4213 $\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\\end{pmatrix}$
$(1,4)(2,3)$ 4321 $\begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\\end{pmatrix}$

### 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:

Family Parameter values Formula for order of a group in the family Proof or justification of formula Evaluation at parameter values Full interpretation of conjugacy class structure
alternating group $A_n$ of degree $n$ degree $n = 4$ $n!/2$ See alternating group, element structure of alternating groups $4!/2 = 4 \cdot 3 \cdot 2 \cdot 1/2 = 12$ #Interpretation as alternating group
projective special linear group of degree two over a finite field of size $q$ $q = 3$, i.e., field:F3, so the group is $PSL(2,3)$ $(q^3 - q)/2 = q(q - 1)(q + 1)/2$ for $q$ odd
$q^3 - q = q(q - 1)(q + 1)$ for $q$ a power of 2
See order formulas for linear groups of degree two, order formulas for linear groups, and projective special linear group of degree two $(3^3 - 3)/2 = (27 - 3)/2 = 12$
Factored version: $3(3 - 1)(3 + 1)/2 = 3(2)(4)/2 = 12$
#Interpretation as projective special linear group of degree two
general affine group of degree one over a finite field of size $q$ field size $q = 4$, i.e., field:F4, so the group is $GA(1,4)$ $q(q - 1)$ See general affine group of degree one $4(4 - 1) = 4(3) = 12$ #Interpretation as general affine group of degree one
von Dyck group with parameters $(p,q,r)$ $(p,q,r) = (3,3,2)$ (note that the order of the parameters is irrelevant, though we usually arrange them in ascending or descending order depending on the convention being followed). $\frac{2}{1/p + 1/q + 1/r - 1}$ See element structure of von Dyck groups $\frac{2}{1/3 + 1/3 + 1/2 - 1} = \frac{2}{1/6} = 12$ #Interpretation as von Dyck group

## Conjugacy class structure

FACTS TO CHECK AGAINST FOR CONJUGACY CLASS SIZES AND STRUCTURE:
Divisibility facts: size of conjugacy class divides order of group | size of conjugacy class divides index of center | size of conjugacy class equals index of centralizer
Bounding facts: size of conjugacy class is bounded by order of derived subgroup
Counting facts: number of conjugacy classes equals number of irreducible representations | class equation of a group

### Interpretation as alternating group

FACTS TO CHECK AGAINST SPECIFICALLY FOR SYMMETRIC GROUPS AND ALTERNATING GROUPS:
Conjugacy class parametrization: cycle type determines conjugacy class (in symmetric group)
Conjugacy class sizes: conjugacy class size formula in symmetric group
Other facts: even permutation (definition) -- the alternating group is the set of even permutations | splitting criterion for conjugacy classes in the alternating group (from symmetric group)| criterion for element of alternating group to be real

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:

Partition Verbal description of cycle type Elements with the cycle type Size of conjugacy class Formula for size Element order
1 + 1 + 1 + 1 four cycles of size one each, i.e., four fixed points $()$ -- the identity element 1 $\! \frac{4!}{(1)^4(4!)}$ 1
2 + 2 double transposition: two cycles of size two $(1,2)(3,4)$, $(1,3)(2,4)$, $(1,4)(2,3)$ 3 $\! \frac{4!}{(2)^2(2!)}$ 2
Total -- $()$, $(1,2)(3,4)$, $(1,3)(2,4)$ and $(1,4)(2,3)$ 4 NA NA

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:

Partition Verbal description of cycle type Elements with the cycle type Combined size of conjugacy classes Formula for combined size Size of each half First split half Second split half Real? Rational? Element order
3 + 1 one 3-cycle, one fixed point $(1,2,3)$, $(1,3,2)$, $(2,3,4)$, $(2,4,3)$, $(3,4,1)$, $(3,1,4)$, $(4,1,2)$, $(4,2,1)$ 8 $\! \frac{4!}{(3)(1)}$ 4 $(1,2,3)$, $(4,2,1)$, $(2,4,3)$, $(3,4,1)$ $(1,3,2)$, $(4,1,2)$, $(2,3,4)$, $(3,1,4)$ No No 3

### 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#Conjugacy class structure

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

Nature of conjugacy class upstairs in $SL_2$ Eigenvalues Characteristic polynomial Minimal polynomial Size of conjugacy class (generic $q$ that is 3 mod 4) Size of conjugacy class ($q = 3$) Number of such conjugacy classes (generic $q$ that is 3 mod 4) Number of such conjugacy classes ($q = 3$) Total number of elements (generic $q$ that is 3 mod 4) Total number of elements ($q = 3$) Representatives as permutations
Diagonalizable over $\mathbb{F}_q$ with equal diagonal entries, hence a scalar $\{ 1,1 \}$ or $\{ -1,-1\}$, both correspond to the same element $(x - a)^2$ where $a \in \{ -1,1 \}$ $x - a$ where $a \in \{ -1,1\}$ 1 1 1 1 1 1 $()$
Diagonalizable over $\mathbb{F}_{q^2}$, not over $\mathbb{F}_q$, eigenvalues square roots of $-1$ Square roots of $-1$ $x^2 + 1$ $x^2 + 1$ $q(q - 1)/2$ 3 1 1 $q(q - 1)/2$ 3 $(1,2)(3,4)$
Not diagonal, has Jordan block of size two $1$ (multiplicity 2) or $-1$ (multiplicity 2). Each conjugacy class has one representative of each type. $(x - a)^2$ where $a \in \{ -1,1 \}$ $x - a$ where $a \in \{ -1,1\}$ $(q^2 - 1)/2$ 4 2 2 $q^2 - 1$ 8 $(1,2,3)$ and $(1,3,2)$
Diagonalizable over $\mathbb{F}_{q^2}$, not over $\mathbb{F}_q$. Must necessarily have no repeated eigenvalues. Eigenvalues not square roots of $-1$. Pair of conjugate elements of $\mathbb{F}_{q^2}$ of norm 1, not square roots of -1. Each pair identified with its negative pair. $x^2 - ax + 1$, $a \ne 0$ irreducible; note that $x^2 - ax + 1$'s pair and $x^2 + ax +1$'s pair get identified. Same as characteristic polynomial $q(q - 1)$ 6 $(q - 3)/4$ 0 $q(q - 1)(q - 3)/4$ 0 --
Diagonalizable over $\mathbb{F}_q$ with distinct (and hence mutually inverse) diagonal entries $\lambda, 1/\lambda$ where $\lambda \in \mathbb{F}_q \setminus \{ 0,1,-1,i,-i \}$ where $i,-i$ are square roots of $-1$. Note that the representative pairs $\{ \lambda, 1/\lambda \}$ and $\{ -\lambda,-1/\lambda \}$ get identified. $x^2 - (\lambda + 1/\lambda)x + 1$, again with identification. $x^2 - (\lambda + 1/\lambda)x + 1$, again with identification. $q(q + 1)$ 12 $(q - 3)/4$ 0 $q(q + 1)(q - 3)/4$ 0 --
Total NA NA NA NA NA $(q + 5)/2$ 4 $(q^3 - q)/2$ 12 NA

### Interpretation as general affine group of degree one

Compare with element structure of general affine group of degree one over a finite field#Conjugacy class structure

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:

Nature of conjugacy class Size of conjugacy class (generic $q$) Size of conjugacy class ($q = 4$) Number of such conjugacy classes (generic $q$) Number of such conjugacy classes ($q = 4$) Total number of elements (generic $q$) Total number of elements ($q = 4$) Representatives of conjugacy classes as permutations
$a = 1, v = 0$ 1 1 1 1 1 1 $()$
$a = 1, v \ne 0$ (conjugacy class is independent of choice of $v$) $q - 1$ 3 1 1 $q - 1$ 3 $(1,2)(3,4)$
$a \ne 1$ (conjugacy class is determined completely by choice of $a$ and is independent of choice of $v$; in other words, each conjugacy class is a coset of the subgroup of translations) $q$ 4 $q - 2$ 2 $q(q - 2)$ 8 $(1,2,3)$ and $(1,3,2)$
Total (--) -- -- $q$ 4 $q(q - 1)$ 12 --

### Interpretation as von Dyck group

alternating group $A_n$ of degree $n$ $n = 4$, i.e., the group $A_4$ (Number of pairs of non-self-conjugate partitions of $n$) + 2(Number of self-conjugate partitions of $n$) See element structure of alternating groups $2 + 2(1) = 4$ #Interpretation as alternating group
projective special linear group of degree two over a finite field of size $q$ $q = 3$, i.e., field:F3, so the group is $PSL(2,3)$ $(q + 5)/2$ for $q$ odd
$q + 1$ for $q$ a power of 2
See element structure of projective special linear group of degree two over a finite field, number of conjugacy classes in projective special linear group of fixed degree over a finite field is PORC function of field size $(3 + 5)/2 = 4$ #Interpretation as projective special linear group of degree two
general affine group of degree one over a finite field of size $q$ field size $q = 4$, i.e., field:F4, so the group is $GA(1,4)$ $q$ See general affine group of degree one $q = 4$ #Interpretation as general affine group of degree one