Alternating group

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

For instance, the alternating group of degree 4 is A₄ = .

Contents

1 Basic properties
2 Conjugacy classes
3 Automorphism group
4 Exceptional isomorphisms
5 Subgroups
6 Schur multipliers

Basic properties

For n > 1, the group A_n is a normal 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 → 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.

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 of cycles of odd length with no two cycles the same length, 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.

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.

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 Z₂². 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 isomorphisms between some of the small alternating groups and small groups of Lie type. These are:

A₄ is isomorphic to the symmetry group of chiral tetrahedral symmetry.
A₅ is isomorphic to PSL₂(4), PSL₂(5), and the symmetry group of chiral icosahedral symmetry.
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 A₁ and A₂ are isomorphic to the trivial group.

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 = A_{4 has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any additional element (except e) generates the whole group.

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 a triple cover. In these cases, then, the Schur multiplier is of order 6.

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