Alternating group

The alternating group $$A_n$$ is the index-two normal subgroup of $$S_n$$ consisting of all even permutations. It is generated by $$3$$ -cycles and is nonabelian simple for $n\geq5$ , giving the first infinite family of finite nonabelian simple groups. The group $$A_5$$ is the smallest nonabelian simple group and is isomorphic to the rotation group of the regular icosahedron. ^{[@isaacs2008]}

Definition

Every permutation is a product of transpositions, and the parity of the number of transpositions is independent of the chosen expression. This defines the sign homomorphism

\operatorname{sgn}:S_n\longrightarrow\{\pm1\}\cong C_2.

For $n\geq2$ , the sign map is surjective, so

|A_n|=\frac{n!}{2}, \qquad A_n\triangleleft S_n, \qquad S_n/A_n\cong C_2.

Generation by 3-cycles

Consequently, a normal subgroup of $$A_n$$ containing one $$3$$ -cycle contains all $$3$$ -cycles and equals $$A_n$$ .

Natural action

$$A_n$$ acts naturally on $\{1,\ldots,n\}$ . For $n\geq4$ it is $$(n-2)$$ -transitive. Given two ordered $$(n-2)$$ -tuples of distinct points, choose a permutation in $$S_n$$ taking one to the other. If it is odd, interchange the two unused letters; this changes parity without affecting the prescribed tuple.

Thus $$A_n$$ is doubly transitive and primitive for $n\geq4$ . A point stabilizer is isomorphic to $A_{n-1}$ , providing the induction structure used in the simplicity proof.

Conjugacy classes in A5

Nonidentity elements of $$A_5$$ have three cycle types: $$3$$ -cycles, products of two disjoint transpositions, and $$5$$ -cycles.

Cycle type	Order	Class size in $$A_5$$
$$(abc)$$	3	20
$$(ab)(cd)$$	2	15
$$(abcde)$$	5	12 and 12

The single $$S_5$$ -class of $$24$$ five-cycles splits into two $$A_5$$ -classes of size $$12$$ .

Simplicity of A5

A Sylow-counting proof is also possible: intersections with Sylow $$5$$ - and $$3$$ -subgroups force a normal subgroup to contain entire conjugacy classes, quickly making its order too large. The class proof exposes the obstruction most directly.

General simplicity theorem

Splitting of conjugacy classes

An $$S_n$$ -conjugacy class of an even permutation splits into two $$A_n$$ -classes exactly when all nontrivial cycle lengths are odd and pairwise distinct. Splitting is equivalent to the full $$S_n$$ -centralizer lying in $$A_n$$ . Repeated cycle lengths permit an odd permutation interchanging two equal cycles; an even-length cycle also supplies an odd centralizing element.

Perfectness and automorphisms

For $n\geq5$ , the derived subgroup $$A_n'$$ is a nontrivial normal subgroup of the nonabelian simple group $$A_n$$ . Hence

\[ A_n'=A_n, \]

so $$A_n$$ is perfect. For $n\geq5$ and $n\neq6$ , every automorphism of $$A_n$$ is induced by conjugation in $$S_n$$ , giving

\operatorname{Aut}(A_n)\cong S_n.

$$A_6$$ has exceptional outer automorphisms, making its outer automorphism group larger than the usual group of order $$2$$ .

Geometry and other realizations

$$A_5$$ is isomorphic to the orientation-preserving rotation group of the regular icosahedron or dodecahedron, and also to $$PSL(2,4)$$ and $$PSL(2,5)$$ . These isomorphisms connect permutation groups, finite-field linear groups, and three-dimensional geometry.

The groups $$A_n$$ form the alternating family in the classification of finite simple groups; the remaining infinite families are groups of Lie type, together with the $$26$$ sporadic groups.

This article follows the permutation-group route of Isaacs, Section 8C, giving the $$A_5$$ class-count base case and a full induction proof. Class splitting, perfectness, and geometric realizations supply the surrounding structure. Terminology was cross-checked against a standard permutation-group text and the cited Chinese and English public encyclopedia revisions. ^{[@isaacs2008]} ^[@dixon1996] ^{[@wikipedia-alternating-zh]} ^{[@wikipedia-alternating-en]}