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In
mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...
, an alternating group is the group of even permutations of a
finite set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The alternating group on a set of ''n'' elements is called the alternating group of degree ''n'', or the alternating group on ''n'' letters and denoted by A''n'' or Alt(''n'').


Basic properties

For , the group A''n'' is the commutator subgroup of the symmetric group S''n'' with
index Index may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a halo megastructure in the ''Halo'' series of ...
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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondit ...
and simple if and only if or . A5 is the smallest non-abelian
simple group Simple or SIMPLE may refer to: *Simplicity Simplicity is the state or quality of being simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something i ...
, having order 60, and the smallest non- solvable group. The group A4 has the Klein four-group V as a proper
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group i ...
, namely the identity and the double transpositions that is the kernel of the surjection of A4 onto . We have the
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules) such that the image An SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teid ...
. 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 decomposition (group theory), 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#Relation with alternating group , Symmetric group''.


Generators and relations

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

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 Klein four-group, the Klein four-group , and is related to Symmetric group#Automorphism group, 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 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)Robinson (1996), [ p. 78] 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 for an indirect isomorphism of using a classification of simple groups of order 60, and Projective linear group#Action on p points, 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'').


Examples ''S''4 and ''A''4


Example A5 as a subgroup of 3-space rotations

A_5 is the group of isometries of a dodecahedron in 3 space, so there is a representation A_5\to SO_3(\mathbb) In this picture the vertices of the polyhedra represent the elements of the group, with the center of the sphere representing the identity element. Each vertex represents a rotation about the axis pointing from the center to that vertex, by an angle equal to the distance from the origin, in radians. Vertices in the same polyhedron are in the same conjugacy class. Since the conjugacy class equation for A_5 is 1+12+12+15+20=60, we obtain four distinct (nontrivial) polyhedra. The vertices of each polyhedron are in bijective correspondence with the elements of its conjugacy class, with the exception of the conjugacy class of (2,2)-cycles, which is represented by an icosidodecahedron on the outer surface, with its antipodal vertices identified with each other. The reason for this redundancy is that the corresponding rotations are by \pi radians, and so can be represented by a vector of length \pi in either of two directions. Thus the class of (2,2)-cycles contains 15 elements, while the icosidodecahedron has 30 vertices. The two conjugacy classes of twelve 5-cycles in A_5 are represented by two icosahedra, of radii 2\pi/5 and 4\pi/5, respectively. The nontrivial outer automorphism in \text(A_5)\simeq Z_2 interchanges these two classes and the corresponding icosahedra.


Example: the 15 puzzle

It can be proved that the 15 puzzle, a famous example of the sliding puzzle, can be represented by the alternating group A_, because the combinations of the 15 puzzle can be generated by Permutation#Definition, 3-cycles. In fact, any 2 \times k - 1 sliding puzzle with square tiles of equal size can be represented by A_.


Subgroups

A4 is the smallest group demonstrating that the converse of Lagrange's theorem (group theory), Lagrange's theorem is not true in general: given a finite group ''G'' and a divisor ''d'' of , there does not necessarily exist a subgroup of ''G'' with order ''d'': the group , of order 12, has no subgroup of order 6. A subgroup of three elements (generated by a cyclic rotation of three objects) with any distinct nontrivial element generates the whole group. For all , A''n'' has no nontrivial (that is, proper)
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup of the group i ...
s. Thus, A''n'' is a
simple group Simple or SIMPLE may refer to: *Simplicity Simplicity is the state or quality of being simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something i ...
for all . A5 is the smallest solvable group, non-solvable group.


Group homology

The group homology of the alternating groups exhibits stabilization, as in stable homotopy theory: for sufficiently large ''n'', it is constant. However, there are some low-dimensional exceptional homology. Note that the Symmetric group#Homology, homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions (additional homology elements).


H1: Abelianization

The first homology group coincides with abelianization, and (since \mathrm_n is perfect group, perfect, except for the cited exceptions) is thus: :H_1(\mathrm_n,\mathrm)=0 for n=0,1,2; :H_1(\mathrm_3,\mathrm)=\mathrm_3^ = \mathrm_3 = \mathrm/3; :H_1(\mathrm_4,\mathrm)=\mathrm_4^ = \mathrm/3; :H_1(\mathrm_n,\mathrm)=0 for n \geq 5. This is easily seen directly, as follows. \mathrm_n is generated by 3-cycles – so the only non-trivial abelianization maps are \mathrm_n \to \mathrm_3, since order 3 elements must map to order 3 elements – and for n \geq 5 all 3-cycles are conjugate, so they must map to the same element in the abelianization, since conjugation is trivial in abelian groups. Thus a 3-cycle like (123) must map to the same element as its inverse (321), but thus must map to the identity, as it must then have order dividing 2 and 3, so the abelianization is trivial. For n < 3, \mathrm_n is trivial, and thus has trivial abelianization. For \mathrm_3 and \mathrm_4 one can compute the abelianization directly, noting that the 3-cycles form two conjugacy classes (rather than all being conjugate) and there are non-trivial maps \mathrm_3 \twoheadrightarrow \mathrm_3 (in fact an isomorphism) and \mathrm_4 \twoheadrightarrow \mathrm_3.


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 also a triple cover. In these cases, then, the Schur multiplier is (the cyclic group) of order 6. These were first computed in . :H_2(\mathrm_n,\mathrm)=0 for n = 1,2,3; :H_2(\mathrm_n,\mathrm)=\mathrm/2 for n = 4,5; :H_2(\mathrm_n,\mathrm)=\mathrm/6 for n = 6,7; :H_2(\mathrm_n,\mathrm)=\mathrm/2 for n \geq 8.


Notes


References

* * *


External links

* * {{DEFAULTSORT:Alternating Group Finite groups Permutation groups