Alternating group

Basic properties

For n > 1, the group A_n is the commutator subgroup of the symmetric group S_n with index 2 and has therefore n! / 2 elements. It is the kernel of the signature group homomorphism sgn : S_n → {1, −1} explained under symmetric group.

The group A_n is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. A₅ is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.

The group A₄ has a Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions { (), (12)(34), (13)(24), (14)(23) }, and maps to A₃ = C₃, from the sequence V → A₄ → A₃ = C₃. In Galois theory, this map, or rather the corresponding map S₄ → S₃, corresponds to associating the Lagrange resolvent cubic to a quartic, which allows the quartic polynomial to be solved by radicals, as established by Lodovico Ferrari.

Conjugacy classes

As in the symmetric group, the conjugacy classes in A_n consist of elements with the same cycle shape. However, if the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type, then there are exactly two conjugacy classes for this cycle shape (Scott 1987, §11.1, p299).

Examples:

Relation with symmetric group

Generators and relations

A_n is generated by 3-cycles, since 3-cycles can be obtained by combining pairs of transpositions. This generating set is often used to prove that A_n is simple for n ≥ 5.

Automorphism group

For n > 3, except for n = 6, the automorphism group of A_n is the symmetric group S_n, with inner automorphism group A_n and outer automorphism group Z₂; the outer automorphism comes from conjugation by an odd permutation.

For n = 1 and 2, the automorphism group is trivial. For n = 3 the automorphism group is Z₂, with trivial inner automorphism group and outer automorphism group Z₂.

The outer automorphism group of A₆ is the Klein four-group V = Z₂ × Z₂, and is related to the outer automorphism of S₆. The extra outer automorphism in A₆ swaps the 3-cycles (like (123)) with elements of shape 3² (like (123)(456)).

Exceptional isomorphisms

There are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are:

More obviously, A₃ is isomorphic to the cyclic group Z₃, and A₀, A₁, and A₂ are isomorphic to the trivial group (which is also SL₁(q) = PSL₁(q) for any q).

Subgroups

A₄ is the smallest group demonstrating that the converse of Lagrange's theorem is not true in general: given a finite group G and a divisor d of | G |, there does not necessarily exist a subgroup of G with order d: the group G = A₄, of order 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any additional element generates the whole group.

For all n ≠ 4, A_n has no nontrivial (i.e., proper) normal subgroups. Thus, A_n is a simple group for all n ≠ 4. A₅ is the smallest non-solvable group.

Group homology

The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sufficiently large n, it is constant. However, there are some low-dimensional exceptional homology. Note that the homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions (additional homology elements).

H1: Abelianization

The first homology group coincides with abelianization, and (since A n is perfect, except for the cited exceptions) is thus:

This is easily seen directly, as follows. A n is generated by 3-cycles – so the only non-trivial abelianization maps are A n → C 3 , since order 3 elements must map to order 3 elements – and for n ≥ 5 all 3-cycles are conjugate, so they must map to the same element in the abelianization, since conjugation is trivial in abelian groups. Thus a 3-cycle like (123) must map to the same element as its inverse (321), but thus must map to the identity, as it must then have order dividing 2 and 3, so the abelianization is trivial.

For n < 3 , A n is trivial, and thus has trivial abelianization. For A 3 and A 4 one can compute the abelianization directly, noting that the 3-cycles form two conjugacy classes (rather than all being conjugate) and there are non-trivial maps A 3 ↠ C 3 (in fact an isomorphism) and A 4 ↠ C 3 .

H2: Schur multipliers

The Schur multipliers of the alternating groups A_n (in the case where n is at least 5) are the cyclic groups of order 2, except in the case where n is either 6 or 7, in which case there is also a triple cover. In these cases, then, the Schur multiplier is (the cyclic group) of order 6. These were first computed in (Schur 1911).