alternating group

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, an alternating group is the
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
of
even permutation In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ord ...
s 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 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 h ...
of the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
S''n'' with
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2 and has therefore ''n''!/2 elements. It is the
kernel Kernel may refer to: Computing * Kernel (operating system) In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
of the signature
group homomorphism Image:Group homomorphism ver.2.svg, 250px, Image of a group homomorphism (h) from G (left) to H (right). The smaller oval inside H is the image of h. N is the Kernel_(algebra)#Group_homomorphisms, kernel of h and aN is a coset of N. In mathematics ...

explained under
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
. The group A''n'' is abelian
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...
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 Johnn ...
if and only if or . A5 is the smallest non-abelian
simple group In mathematics, a simple group is a nontrivial Group (mathematics), group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgrou ...
, having order 60, and the smallest non-
solvable group 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 group A4 has the
Klein four-group 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). I ...
V as a proper
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, namely the identity and the double transpositions that is the kernel of the surjection of A4 onto . We have the
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. In
Galois theory 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 ...
, 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 ratio ...
cubic to a quartic, which allows the
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to be solved by radicals, as established by
Lodovico Ferrari Lodovico de Ferrari (2 February 1522 – 5 October 1565) was an Italians, Italian mathematician. Biography Born in Bologna, Lodovico's grandfather, Bartolomeo Ferrari, was forced out of Milan to Bologna. Lodovico settled in Bologna, and he be ...
.

Conjugacy classes

As in the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...
, 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 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'').

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 $A_5\to SO_3\left(\mathbb\right)$ 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 $\text\left(A_5\right)\simeq Z_2$ 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 In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
s. Thus, A''n'' is a
simple group In mathematics, a simple group is a nontrivial Group (mathematics), group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgrou ...
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: :$H_1\left(\mathrm_n,\mathrm\right)=0$ for $n=0,1,2$; :$H_1\left(\mathrm_3,\mathrm\right)=\mathrm_3^ = \mathrm_3 = \mathrm/3$; :$H_1\left(\mathrm_4,\mathrm\right)=\mathrm_4^ = \mathrm/3$; :$H_1\left(\mathrm_n,\mathrm\right)=0$ 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 . :$H_2\left(\mathrm_n,\mathrm\right)=0$ for $n = 1,2,3$; :$H_2\left(\mathrm_n,\mathrm\right)=\mathrm/2$ for $n = 4,5$; :$H_2\left(\mathrm_n,\mathrm\right)=\mathrm/6$ for $n = 6,7$; :$H_2\left(\mathrm_n,\mathrm\right)=\mathrm/2$ for $n \geq 8$.

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