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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 $\operatorname{sgn}(\sigma)=(-1)^{n-c(\sigma)}$ 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 $n\ge2$ , $2\,|A_n|=n!$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 67 results over 18 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

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

A4A_4A4​ consists of the identity, eight 333-cycles, and three products of disjoint transpositions