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 {1,...,n} 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 = {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 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 double transpositions {(12)(34), (13)(24), (14)(23)}, and maps to A₃ = C₃, form the 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, 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:

• 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	Aut(An)	Out(An)

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

• A₄ is isomorphic to PSL₂(3) and 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 (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.

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:

for n = 1,2;

;

;

for .

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

for n = 1,2,3;

for n = 4,5;

for n = 6,7;

for .

References

This article needs additional citations for verification. Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (January 2008)

• Scott, W.R. (1987), Group Theory, New York: Dover Publications, ISBN 978-0-486-65377-8