alternating group
   HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an alternating group is the group of even permutations of a
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ...
. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted by or


Basic properties

For , the group A''n'' is the commutator subgroup of the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
S''n'' with
index Index (: indexes or indices) 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 the Halo Array in the ...
2 and has therefore ''n''!/2 elements. It is the kernel of the signature
group homomorphism In mathematics, given two groups, (''G'',∗) and (''H'', ·), a group homomorphism from (''G'',∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) whe ...
explained under
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
. The group A''n'' is abelian
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
and
simple Simple or SIMPLE may refer to: *Simplicity, the state or quality of being simple Arts and entertainment * ''Simple'' (album), by Andy Yorke, 2008, and its title track * "Simple" (Florida Georgia Line song), 2018 * "Simple", a song by John ...
if and only if or . A5 is the smallest non-abelian simple group, having order 60, and thus the smallest non- solvable group. The group A4 has the
Klein four-group In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...
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 N of the group ...
, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto . We have the exact sequence . In
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, this map, or rather the corresponding map , corresponds to associating the
Lagrange resolvent In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rat ...
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 In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
, any two elements of A''n'' that are conjugate by an element of A''n'' must have the same 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
permutation In mathematics, a permutation of a set can mean one of two different things: * an arrangement of its members in a sequence or linear order, or * the act or process of changing the linear order of an ordered set. An example of the first mean ...
s (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''. As finite symmetric groups are the groups of all permutations of a set with finite elements, and the alternating groups are groups of even permutations, alternating groups are subgroups of finite symmetric groups.


Generators and relations

For ''n'' ≥ 3, 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 the Klein four-group , 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 ).


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/ref> and the symmetry group of chiral tetrahedral symmetry. * A5 is isomorphic to PSL2(4), PSL2(5), and the symmetry group of chiral
icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
. (See for an indirect isomorphism of using a classification of simple groups of order 60, and 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 S4 and A4


Example A5 as a subgroup of 3-space rotations

A5 is the group of isometries of a dodecahedron in 3-space, so there is a representation . 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 A5 is , 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 radians, and so can be represented by a vector of length 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 A5 are represented by two icosahedra, of radii 2/5 and 4/5, respectively. The nontrivial outer automorphism in 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 A15, because the combinations of the 15 puzzle can be generated by 3-cycles. In fact, any sliding puzzle with square tiles of equal size can be represented by A2''k''−1.


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 , 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 N of the group ...
s. Thus, A''n'' is a simple group for all . A5 is the smallest 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 homology of the symmetric group exhibits similar stabilization, but without the low-dimensional exceptions (additional homology elements).


''H''1: Abelianization

The first
homology group In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...
coincides with abelianization, and (since A''n'' is perfect, except for the cited exceptions) is thus: :''H''1(A''n'', Z) = Z1 for ''n'' = 0, 1, 2; :''H''1(A3, Z) = A = A3 = Z3; :''H''1(A4, Z) = A = Z3; :''H''1(A''n'', Z) = Z1 for ''n'' ≥ 5. This is easily seen directly, as follows. A''n'' is generated by 3-cycles – so the only non-trivial abelianization maps are since order-3 elements must map to order-3 elements – and for 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 , A''n'' is trivial, and thus has trivial abelianization. For A3 and A4 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 (in fact an isomorphism) and .


''H''2: 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(A''n'', Z) = Z1 for ''n'' = 1, 2, 3; :''H''2(A''n'', Z) = Z2 for ''n'' = 4, 5; :''H''2(A''n'', Z) = Z6 for ''n'' = 6, 7; :''H''2(A''n'', Z) = Z2 for ''n'' ≥ 8.


Notes


References

* * *


External links

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