Definitions

alternating-group

Alternating group

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on the set {1,...,n} is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt(n).

For instance, the alternating group of degree 4 is A4 = {e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23)} (see cycle notation).

Basic properties

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

The group An 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 An 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 .

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

n mbox{Aut}(A_n) mbox{Out}(A_n)
ngeq 4, nneq 6 S_n, C_2,
n=1,2, 1, 1,
n=3, C_2, C_2,
n=6, S_6 rtimes C_2 V=C_2 times C_2

For n > 3, except for n = 6, the automorphism group of An is the symmetric group Sn, with inner automorphism group An 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 isomorphisms between some of the small alternating groups and small groups of Lie type. These are:

More obviously, A3 is isomorphic to the cyclic group Z3, and 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 (except e) generates the whole group.

Group homology

The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sufficiently large n, it is constant.

H1: Abelianization

The first homology group coincides with abelianization, and (since A_n is perfect, except for the cited exceptions) is thus:
H_1(A_3,mathbf{Z})=A_3^{text{ab}} = A_3 = mathbf{Z}/3;
H_1(A_4,mathbf{Z})=A_4^{text{ab}} = mathbf{Z}/3;
H_1(A_n,mathbf{Z})=0 for n=1,2 and ngeq 5.

H2: Schur multipliers

The Schur multipliers of the alternating groups An (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.

H_2(A_n,mathbf{Z})=0 for n = 1,2,3;
H_2(A_n,mathbf{Z})=mathbf{Z}/6 for n = 6,7;
H_2(A_n,mathbf{Z})=mathbf{Z}/2 for n = 4,5 and n geq 8.

References

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