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$A_{4}$ has a normal Klein four subgroup and four conjugacy classes

Example

Let $V_{4} = {1, (12) (34), (13) (24), (14) (23)} \subseteq A_{4}$ . Then $V_{4}$ is a normal subgroup of $A_{4}$ of order $4$ , the quotient $A_{4} / V_{4}$ has order $3$ , and $A_{4}$ has four conjugacy classes, of sizes $1$ , $3$ , $4$ , and $4$ .

Facts & Assumptions

Given: The alternating group $A_{4} \subseteq S_{4}$ and the subset $V_{4} = {1, (12) (34), (13) (24), (14) (23)}$ .

[F1]

Two permutations of $S_{n}$ are conjugate exactly when they have the same cycle type (Two elements of $S_{n}$ are conjugate if and only if they have the same cycle type).

[F2]

For $n \geq 2$ , the $S_{n}$ -class of an element of $A_{n}$ splits into two $A_{n}$ -classes of equal size exactly when all cycle lengths, $1$ -cycles included, are odd and pairwise distinct. (For $n \geq 2$ , an $S_{n}$ -class of an even permutation splits in $A_{n}$ exactly when all cycle lengths, including $1$ -cycles, are odd and distinct)

[F3]

For finite $G$ and $N ⊴ G$ , $∣ G / N ∣ = ∣ G ∣ / ∣ N ∣$ (If $[G : N]$ is finite then $∣ G / N ∣ = [G : N]$ ; for finite $G$ this equals $∣ G ∣ / ∣ N ∣$ ).

[A1]

The product of two distinct double transpositions on four letters is the remaining double transposition, and a product of two transpositions is an even permutation; $∣ A_{4} ∣ = 12$ .

[A2]

Conjugating a double transposition $(a b) (c d)$ by a permutation $σ$ of the four letters gives the double transposition $(σ (a) σ (b)) (σ (c) σ (d))$ .

Verification

technique · direct

1.1

By [A1] the three nonidentity elements of $V_{4}$ are even permutations and multiply pairwise to the third, so $V_{4}$ is a subgroup of $A_{4}$ of order $4$ .

A1given

1.2

For the $3$ -cycle $(123)$ , the cycle lengths are $3$ and $1$ , odd and distinct, so [F2] applies: its $S_{4}$ -class, of size $8$ , splits into two $A_{4}$ -classes of equal size $4$ . These two classes account for all eight $3$ -cycles of $A_{4}$ .

F2givenalgebra

2.1

By [A2], any conjugate of a double transposition is again a double transposition, and there are exactly three of them; by [F1] they all lie in the single $S_{4}$ -class of cycle type $(2, 2)$ . Hence every conjugate of every element of $V_{4}$ lies in $V_{4}$ , so $V_{4} ⊴ A_{4}$ .

F1A2step 1.1given

2.2

By [F3], $∣ A_{4} / V_{4} ∣ = ∣ A_{4} ∣ / ∣ V_{4} ∣ = 12 / 4 = 3$ .

F3step 1.1A1

3.1

The identity class of $A_{4}$ is ${1}$ . For the double transposition $(12) (34)$ , the cycle type $(2, 2)$ has two equal lengths, so [F2] does not apply and its $S_{4}$ -class does not split: its $A_{4}$ -class is the whole class of double transpositions, of size $3$ by step 2.1.

F2step 2.1given

4.1

Every nonidentity element of $A_{4}$ is a double transposition or a $3$ -cycle, so steps 3.1 and 1.2 list all classes: sizes $1$ , $3$ , $4$ , $4$ .

step 3.1step 1.2algebra∎

Depends on

Used by

The character table of A₄ Example

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Pavel Etingof et al., Introduction to Representation Theory, Section 3.5 (standard reference, not scraped)

A4A4​ has a normal Klein four subgroup and four conjugacy classes