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$A_{4}$ consists of the identity, eight $3$ -cycles, and three products of disjoint transpositions

Example

The alternating group $A_{4}$ consists of

the identity;
the eight three-cycles $(0 1 2)$ , $(0 2 1)$ , $(0 1 3)$ , $(0 3 1)$ , $(0 2 3)$ , $(0 3 2)$ , $(1 2 3)$ , and $(1 3 2)$ ;
the three products $(0 1) (2 3)$ , $(0 2) (1 3)$ , and $(0 3) (1 2)$ .

Facts & Assumptions

Given: The symmetric group $S_{4}$ and its alternating subgroup $A_{4}$ .

[L1]

Every permutation has a unique disjoint-cycle type, and a cycle of length $k$ has sign $(- 1)^{k - 1}$ (Every permutation of a finite set is a product of pairwise disjoint cycles, uniquely up to reordering and cyclic rotation, A $k$ -cycle has sign $(- 1)^{k - 1}$ , and $sgn (σ) = (- 1)^{n - c (σ)}$ when fixed points are counted as cycles).

[L2]

The alternating group has $4! / 2 = 12$ elements in degree $4$ ( $A_{n}$ is normal in $S_{n}$ ; for $n \geq 2$ , $2 ∣ A_{n} ∣ = n!$ , while $A_{n} = S_{n}$ for $n = 0, 1$ ).

Verification

technique · direct

1.1

The possible cycle types in $S_{4}$ are the identity; one transposition; two disjoint transpositions; a three-cycle with one fixed point; and a four-cycle. By [L1], exactly the identity, the three-cycles, and the products of two disjoint transpositions are even. There are two orientations on each of the four three-point supports and three partitions into two unordered pairs, giving exactly the displayed list.

givenL1

2.1

The list has $1 + 8 + 3 = 12$ elements, and [L2] gives $∣ A_{4} ∣ = 4! / 2 = 12$ , so it contains every even permutation and no other element.

step 1.1L1L2∎

Depends on

Used by

A₄ has no subgroup of order 6 Example

Dependency tree · two levels

18 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

T. W. Judson, Abstract Algebra: Theory and Applications, §5.1, Example 8 (standard reference, not scraped)

A4A4​ consists of the identity, eight 33-cycles, and three products of disjoint transpositions