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In mathematics, an alternating group is the group of even permutations of a finite set. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, the Permutations of a Finite set (ie the bijective mappings from the set to itself fall into two classes of equal size the even In Mathematics, a set is called finite if there is a Bijection between the set and some set of the form {1 2. 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). In combinatorial Mathematics, the cycle notation is a useful convention for writing down a Permutation in terms of its constituent cycles

1 Basic properties
2 Conjugacy classes
3 Automorphism group
4 Exceptional isomorphisms
5 Subgroups
6 Group homology
- 6.1 H₁: Abelianization
- 6.2 H₂: Schur multipliers
7 References

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. In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH Definition The factorial function is formally defined by n!=\prod_{k=1}^n k It is the kernel of the signature group homomorphism sgn : S_n → {1, −1} explained under symmetric group. 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 In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying

The group A_n is abelian if and only if n ≤ 3 and simple if and only if n = 3 or n ≥ 5. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the ↔ SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning A₅ is the smallest non-abelian simple group, having order 60, and the smallest non-solvable group. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally

Conjugacy classes

As in the symmetric group, the conjugacy classes in A_n consist of elements with the same cycle shape. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class 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. In several fields of Mathematics the term permutation is used with different but closely related meanings

Automorphism group

For more details on this topic, see Automorphisms of the symmetric and alternating groups. In Group theory, a branch of Mathematics, the Automorphisms and Outer automorphisms of the Symmetric groups and Alternating groups are

$n$	$Aut(A n)$	$Out(A n)$
$n\geq 4, n\neq 6$	$S_n\,$	$C_2\,$
$n=1,2\,$	$1\,$	$1\,$
$n=3\,$	$C_2\,$	$C_2\,$
$n=6\,$	$S_6 \rtimes C_2$	$V=C_2 \times C_2$

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. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner

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. In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying 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. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, a group of Lie type G(k is a (not necessarily finite group of rational points of a reductive Linear algebraic group G with These are:

A₄ is isomorphic to PSL₂(3) and the symmetry group of chiral tetrahedral symmetry. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is A regular Tetrahedron has 12 rotational (or orientation-preserving 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. A regular Icosahedron has 60 rotational (or orientation-preserving 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 Z₃, and A₁ and A₂ are isomorphic to the trivial group (which is also SL₁(q)=PSL₁(q) for any q). In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, a trivial group is a group consisting of a single element

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. Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of 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: for sufficiently large n, it is constant. In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper In Mathematics, stable homotopy theory is that part of Homotopy theory (and thus Algebraic topology) concerned with all structure and phenomena that remain

H₁: Abelianization

The first homology group coincides with abelianization, and (since $A n$ is perfect, except for the cited exceptions) is thus:

$H_1(A_3,\mathbf{Z})=A_3^{\text{ab}} = A_3 = \mathbf{Z}/3$ ;

$H_1(A_4,\mathbf{Z})=A_4^{\text{ab}} = \mathbf{Z}/3$ ;

$H_1(A_n,\mathbf{Z})=0$ for

n = 1,2

and $n\geq 5$ . In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup 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

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 Mathematics, more specifically in Group theory, the Schur multiplier is an important invariant of a group that has applications in many areas of mathematics In these cases, then, the Schur multiplier is of order 6.

$H_2(A_n,\mathbf{Z})=0$ for

n = 1,2,3

;

$H_2(A_n,\mathbf{Z})=\mathbf{Z}/6$ for

n = 6,7

;

$H_2(A_n,\mathbf{Z})=\mathbf{Z}/2$ for

n = 4,5

and $n \geq 8$ .

References

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

Dover Publications is an American book Publisher founded in 1941 by Hayward Cirker and his wife Blanche

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