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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an alternating group is the group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of even permutations of a finite set
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
. The alternating group on the set 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 = (see cycle notation
Cycle notation

In combinatorics mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycle s....
).

Basic properties
For n > 1, the group An is the commutator subgroup
Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generating set of a group by all the commutators of the group....
 of the symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
 Sn with index 2 and has therefore n!
Factorial

In mathematics, the factorial of a negative and non-negative numbers integer n, denoted by n!, is the Product of all positive integers less than or equal to n....
/2 elements.






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In mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
, an alternating group is the group
Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an Binary operation that combines any two of its element to form a third element....
 of even permutations of a finite set
Finite set

In mathematics, finite set is a Set that has a finite number of element . For example,is a finite set with five elements. The number of elements of a finite set is a natural number , and is called the cardinality of the set....
. The alternating group on the set 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 = (see cycle notation
Cycle notation

In combinatorics mathematics, the cycle notation is a useful convention for writing down a permutation in terms of its constituent cycle s....
).

Basic properties


For n > 1, the group An is the commutator subgroup
Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generating set of a group by all the commutators of the group....
 of the symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
 Sn with index 2 and has therefore n!
Factorial

In mathematics, the factorial of a negative and non-negative numbers integer n, denoted by n!, is the Product of all positive integers less than or equal to n....
/2 elements. It is the kernel
Kernel (algebra)

In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective....
 of the signature group homomorphism
Group homomorphism

In mathematics, given two group and , a group homomorphism from to is a function h : G ? H such that for all u and v in G it holds that...
 sgn : Sn ? explained under symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
.

The group An is abelian
Abelian group

An abelian group, also called a commutative group, is a group satisfying the requirement that the product of elements does not depend on their order ....
 if and only if
If and only if

If and only if, in logic and fields that rely on it such as mathematics and philosophy, is a biconditional logical connective between statements....
 n = 3 and simple
Simple group

In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself....
 if and only if n = 3 or n = 5. A5 is the smallest non-abelian simple group
Simple group

In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself....
, having order 60, and the smallest non-solvable group
Solvable group

In the history of mathematics, the origins of group theory lie in the search for a Mathematical_proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory....
.

Conjugacy classes


As in the symmetric group
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
, the conjugacy classes
Conjugacy class

In mathematics, especially group theory, the elements of any group may be partition of a set into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure....
 in An consist of elements with the same cycle shape
Cycle decomposition

A cycle decomposition in mathematics may refer to one of two related concepts.* In graph theory, a cycle decomposition is a partitioning of the vertices of a graph into subsets, such that the vertices in each subset lie on a Cycle ....
. 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
    Permutation

    In several fields of mathematics the term permutation is used with different but closely related meanings. They all relate to the notion of mapping the element s of a set to other elements of the same set, i.e., exchanging elements of a set....
    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.


Automorphism group



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
Outer automorphism group

In mathematics, the outer automorphism group of a group Gis the quotient group Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms....
 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
Klein four-group

In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of Order 2 ....
 V = Z2 × Z2, and is related to the outer automorphism of S6
Symmetric group

In mathematics, the symmetric group on a Set X, denoted by SX, or Sym, is the group whose underlying set is the set of all bijective function s from X to X, in which the group operation is that of Function composition, i.e., two such functions f and g can be composed to yield a new bijective function ,...
. 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 isomorphism
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
s 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
    Symmetry group

    The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
     of chiral tetrahedral symmetry
    Tetrahedral symmetry

    A regular tetrahedron has 12 rotational symmetries, and a total of 24 symmetries including transformations that combine a reflection and a rotation....
    .
  • A5 is isomorphic to PSL2(4), PSL2(5), and the symmetry group of chiral icosahedral symmetry
    Icosahedral symmetry

    File:Soccer ball.svgA regular icosahedron has 60 rotational symmetries, and a total of 120 symmetries including transformations that combine a reflection and a rotation....
    .
  • A6 is isomorphic to PSL2(9) and PSp4(2)'
  • A8 is isomorphic to PSL4(2)


More obviously, A3 is isomorphic to the cyclic group
Cyclic group

In group theory, a cyclic group or monogenous group is a group that can be generating set of a group by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g ....
 Z3, and A1 and A2 are isomorphic to the trivial group
Trivial group

In mathematics, a trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of 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
Lagrange's theorem (group theory)

Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G....
 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
Stable homotopy theory

In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor....
: for sufficiently large n, it is constant.

H1: Abelianization

The first homology group coincides with abelianization, and (since is perfect
Perfect group

In mathematics, in the realm of group theory, a Group is said to be perfect if it equals its own commutator subgroup, or equivalently, if the group has no nontrivial abelian group quotient group....
, except for the cited exceptions) is thus: ; ; for and .

H2: Schur multipliers


The Schur multiplier
Schur multiplier

In mathematics, more specifically in group theory, the Schur multiplier is an important invariant of a group that has applications in many areas of mathematics....
s 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 ; for ; for and .