Alternating group explained

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or

Basic properties

For, 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 explained under symmetric group.

The group A_n is abelian if and only if and simple if and only if or . A₅ is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group.

The group A₄ has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions, that is the kernel of the surjection of A₄ onto . We have the exact sequence . In Galois theory, this map, or rather the corresponding map, 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, any two elements of A_n that are conjugate by an element of A_n must have the same cycle shape. The converse is not necessarily true, 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 .

Examples:

• The two permutations (123) and (132) are not conjugates in A₃, although they have the same cycle shape, and are therefore conjugate in S₃.
• The permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A₈, although the two permutations have the same cycle shape, so they are conjugate in S₈.

Relation with symmetric group

See Symmetric group.As finite symmetric groups are the groups of all permutations of a set with finite elements, and the alternating groups are groups of even permutations, alternating groups are subgroups of finite symmetric groups.

Generators and relations

For n ≥ 3, 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 .

Automorphism group

n	Aut(An)	Out(An)
n ≥ 4, n ≠ 6	Sn	Z2
n = 1, 2	Z1	Z1
n = 3	Z2	Z2
n = 6	S6 ⋊ Z2	V = Z2 × Z2

For, except for, 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 and 2, the automorphism group is trivial. For 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, 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 ).

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:

• A₄ is isomorphic to PSL₂(3)^[1] and the symmetry group of chiral tetrahedral symmetry.
• A₅ is isomorphic to PSL₂(4), PSL₂(5), and the symmetry group of chiral icosahedral symmetry. (See^[1] for an indirect isomorphism of using a classification of simple groups of order 60, and here for a direct proof).
• A₆ is isomorphic to PSL₂(9) and PSp₄(2)'.
• A₈ is isomorphic to PSL₄(2).

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

Notes and References

1. Robinson (1996), [{{Google books|plainurl=y|id=lqyCjUFY6WAC|page=78|text=PSL}} p. 78]