| Dictionary: alternating group |
| 5min Related Video: alternating group |
| Wikipedia: Alternating group |
| Group theory | ||||||||
Group theory
|
||||||||
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).
Contents |
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.
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.
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 (Scott 1987, §11.1, p299).
Examples:
| n | Aut(An) | Out(An) |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
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)).
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).
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.
The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sufficiently large n, it is constant.
The first homology group coincides with abelianization, and (since An is perfect, except for the cited exceptions) is thus:
for n = 1,2;
;
;
for
.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.
for n = 1,2,3;
for n = 4,5;
for n = 6,7;
for
.| 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) |
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
| Best of the Web: alternating group |
Some good "alternating group" pages on the web:
Math mathworld.wolfram.com |
| alternating-group flashing light (navigation) | |
| Papa Gato (1986 Album by Poncho Sanchez) | |
| Marina Rosenfeld (Electronica Artist, '90s, 2000s) |
| Are eggs part of the meat and alternative group from canadas food guide? | |
| What 90's alternative group had the most number one hits? | |
| The backbone of the DNA molecule is alternating and phosphate groups? |
Copyrights:
![]() | Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read more | |
![]() | Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Alternating group". Read more |
Mentioned in