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Alternating group

In group theory, the alternating group A_n (for n \geq 2) is defined as the kernel of the sign homomorphism \epsilon: S_n \to \{ \pm 1 \}, where S_n is the symmetric group on n letters, and thus consists precisely of the even permutations in S_n.^[1] This subgroup has order |A_n| = n!/2 and index [S_n : A_n] = 2, making it the unique normal subgroup of S_n of that index.^[1] As the commutator subgroup of S_n, A_n plays a central role in the structure of permutation groups and serves as a foundational example of non-abelian simple groups for n \geq 5.^[1] Key properties of A_n include its normality in S_n, established by the fact that conjugation preserves the sign of a permutation.^[2] For n \neq 4, A_n is a simple group, meaning it has no nontrivial normal subgroups, which underscores its importance in the classification of finite simple groups; however, A_4 is not simple, as it contains the normal Klein four-group as a subgroup.^[1] Additionally, A_n is generated by all 3-cycles for n \geq 3, and its derived subgroup A_n' coincides with A_n itself for n \geq 5, reflecting its perfect nature in those cases.^[1] For small values of n, the structure of A_n varies notably: A_2 is trivial, A_3 is cyclic of order 3 (isomorphic to \mathbb{Z}/3\mathbb{Z}), and A_4 has order 12.^[1] Subgroups of index n in A_n (for n \neq 4) are isomorphic to A_{n-1}, highlighting recursive structural similarities.^[1] These groups are indispensable in modern algebra, appearing in applications from Galois theory to the study of solvability and representation theory.^[2]

Definition and Fundamentals

Definition

In group theory, the alternating group A_n is the subgroup of the symmetric group S_n, which consists of all permutations of n elements, comprising precisely those permutations that are even.^[3] An even permutation is defined as one that can be expressed as a product of an even number of transpositions (2-cycles).^[4] Formally, A_n is the kernel of the sign homomorphism \sgn: S_n \to \{ \pm 1 \}, where the sign of a permutation \sigma is \sgn(\sigma) = 1 if \sigma is even and \sgn(\sigma) = -1 if \sigma is odd; thus,

A_n = \{ \sigma \in S_n \mid \sgn(\sigma) = 1 \}.

^[5] This kernel property establishes A_n as a normal subgroup of S_n of index 2.^[6] The standard notation A_n (or sometimes \Alt(n)) is used for n \geq 3, as these are the cases of primary interest in permutation group theory; for completeness, A_1 and A_2 are both the trivial group consisting solely of the identity permutation.^[7] The concept and terminology of the alternating group originated in the work of Camille Jordan, who introduced the term in his 1873 paper on transitive groups.^[8]

Basic Properties

The alternating group A_n on n letters, for n \geq 2, consists of all even permutations in the symmetric group S_n and has order |A_n| = n!/2. This cardinality follows from the surjectivity of the sign homomorphism \operatorname{sgn}: S_n \to \{ \pm 1 \}, whose kernel is precisely A_n.^[9]^[10] A permutation is even if and only if it can be expressed as a product of an even number of transpositions, and this parity is independent of the particular decomposition into transpositions. The sign function provides the rigorous parity argument: for any permutation \sigma, \operatorname{sgn}(\sigma) = (-1)^k where k is the number of transpositions in any decomposition of \sigma, ensuring the even permutations form a well-defined subgroup of index 2 in S_n.^[10] For n \geq 3, the alternating group A_n is generated by the set of all 3-cycles. To see this, note that any even permutation is a product of an even number of transpositions, and a product of two disjoint transpositions such as (ab)(cd) equals (acb)(acd), a product of two 3-cycles; more generally, any even permutation reduces to the identity via conjugation and multiplication by 3-cycles. In particular, A_n is generated by the 3-cycle (1\,2\,3) and its conjugates under S_n./10:_Normal_Subgroups_and_Factor_Groups/10.02:_The_Simplicity_of_the_Alternating_Groups)^[11] The center of A_n is trivial, Z(A_n) = \{ e \}, for all n \geq 4. This holds because any non-identity even permutation fails to commute with some 3-cycle in A_n, as conjugation by elements of S_n splits cycles in a way that disrupts centrality within the even permutations.^[12] The derived subgroup (commutator subgroup) of A_n equals A_n itself for n \geq 5, making A_n a perfect group in these cases. For n = 4, the derived subgroup is the Klein four-group consisting of the identity and the three double transpositions.^[12]

Relation to Symmetric Group

Subgroup Structure

The alternating group A_n is a normal subgroup of the symmetric group S_n for n \geq 2, as it forms the kernel of the sign homomorphism \operatorname{sgn}: S_n \to \{\pm [1](/page/1)\}, which maps even permutations to +[1](/page/1) and odd permutations to -[1](/page/1).^[3] This kernel property directly implies normality, since the kernel of any group homomorphism is a normal subgroup of the domain.^[13] The index of A_n in S_n is 2, denoted [S_n : A_n] = 2, reflecting that S_n partitions into exactly two cosets of A_n.^[14] Consequently, the order of A_n is n!/2, half the order of S_n, by Lagrange's theorem.^[15] The coset decomposition is S_n = A_n \sqcup \tau A_n, where \tau is any fixed odd permutation (such as the transposition (1\, 2)), and the second coset consists precisely of all odd permutations in S_n.^[14] By the first isomorphism theorem, the quotient group S_n / A_n is isomorphic to \mathbb{Z}/2\mathbb{Z}, the cyclic group of order 2, as the sign homomorphism is surjective onto \{\pm 1\} \cong \mathbb{Z}/2\mathbb{Z}.^[15] This quotient captures the parity distinction between even and odd permutations. Furthermore, S_n acts on the set of its two cosets \{A_n, \tau A_n\} by left multiplication, yielding a transitive action that induces the sign homomorphism and underscores the structural role of A_n as the unique subgroup of index 2 in S_n.^[14]

Conjugacy Classes

The conjugacy classes of the alternating group A_n consist of the even permutations in S_n, which are precisely those whose cycle decompositions contain an even number of even-length cycles.^[16] In S_n, conjugacy classes are partitioned solely by cycle type, but in the subgroup A_n, even permutations of certain cycle types may form a single class, while others split into two equal-sized classes under conjugation by even permutations.^[17] A conjugacy class of an even permutation in S_n splits into two classes in A_n if and only if its cycle type consists of disjoint cycles of distinct odd lengths (including at most one fixed point, as multiple fixed points would repeat the length-1 cycle).^[18] In such cases, no odd permutation commutes with a representative of the class, so the centralizer C_{S_n}(\sigma) lies entirely within A_n, causing the S_n-class to divide evenly under the index-2 subgroup. For cycle types with at least one even-length cycle or repeated odd lengths, the centralizer in S_n contains odd permutations, and the A_n-class remains unsplit with size equal to that of the corresponding S_n-class.^[18] For example, double transpositions of cycle type (2,2), such as (1\,2)(3\,4), involve even-length cycles and thus do not split; in A_4, the three such elements form a single conjugacy class of size 3.^[17] In contrast, a 5-cycle in A_5 has cycle type (5) (a single odd length with no fixed points), so its S_5-class of 24 elements splits into two A_5-classes of size 12 each.^[19] The size of any conjugacy class \mathrm{cl}(\sigma) in A_n is given by the class equation |\mathrm{cl}(\sigma)| = |A_n| / |C_{A_n}(\sigma)|, where C_{A_n}(\sigma) = C_{S_n}(\sigma) \cap A_n is the centralizer of \sigma in A_n.^[17] For 3-cycles, which have cycle type (3,1^{n-3}), the presence of repeated length-1 cycles (for n \geq 5) prevents splitting, and all such elements are conjugate in A_n. The total number of 3-cycles in A_n (for n \geq 3) is \binom{n}{3} \times 2, obtained by choosing 3 elements and forming either orientation of the cycle.^[19] For a representative 3-cycle \sigma with n \geq 5, the centralizer in S_n has order $3(n-3)!, half of which is even, so |C_{A_n}(\sigma)| = \frac{3(n-3)!}{2} and the class size is \frac{n!/2}{3(n-3)!/2} = \frac{n(n-1)(n-2)}{3} = \binom{n}{3} \times 2.^[17]

Presentation and Automorphisms

Generators and Relations

The alternating group A_n for n \geq 3 is generated by its 3-cycles; every even permutation can be expressed as a product of these elements, since any product of two disjoint transpositions is a product of two 3-cycles, and any even permutation is a product of an even number of transpositions.^[12] Although the full set of 3-cycles is generally required, A_n admits a minimal generating set of size two for all n > 2 except n = 6, 7, where three generators are needed.^[20] For n = 4, two 3-cycles suffice: A_4 = \langle (1\, 2\, 3), (1\, 2\, 4) \rangle. This group has the presentation

\langle x, y \mid x^3 = y^3 = (xy)^2 = 1 \rangle,

where x = (1\, 2\, 3) and y = (1\, 2\, 4); the relations ensure the group order is 12, matching |A_4|.^[21] For n \geq 5, A_n is generated by the 3-cycles, and a presentation can be derived from the Coxeter presentation of the symmetric group S_n using the Reidemeister-Schreier rewriting process to account for the index-2 subgroup of even permutations.^[20] Explicit 2-generator presentations exist with O(\log n) relations of total length O(\log^2 n); for example, when n is odd, suitable generators are the n-cycle y = (1\, 2\, \dots \, n) and xyx where x = (1\, 2).^[20]

Automorphism Group

The inner automorphism group of the alternating group A_n for n \geq 4 is isomorphic to A_n itself, as the center Z(A_n) is trivial in these cases, yielding \operatorname{Inn}(A_n) \cong A_n / Z(A_n) \cong A_n.^[22] For n \geq 4 and n \neq 6, the full automorphism group \operatorname{Aut}(A_n) is isomorphic to the symmetric group S_n. This isomorphism arises from the action of S_n on A_n by conjugation, which induces all automorphisms of A_n; specifically, the natural homomorphism S_n \to \operatorname{Aut}(A_n) has trivial kernel, hence is an isomorphism. Conjugation by even permutations yields the inner automorphisms \operatorname{Inn}(A_n) \cong A_n of index 2, while conjugation by odd permutations provides the nontrivial outer automorphisms, with \operatorname{Out}(A_n) \cong C_2. Thus, |\operatorname{Aut}(A_n)| = n!. For example, conjugation by an odd permutation in S_n provides the nontrivial outer automorphism of A_n.^[22] The case n = 6 is exceptional, where \operatorname{Aut}(A_6) has order $2 \cdot |S_6| = [1440](/page/1440), so \operatorname{Out}(A_6) \cong C_2 \times C_2. This enlargement stems from an additional outer automorphism of A_6, induced by the nontrivial outer automorphism of S_6, which itself arises from the enlarged automorphism group of the line graph of the complete graph K_6 (isomorphic to S_6 \times C_2, unlike for other n).^[23]

Isomorphisms and Examples

Exceptional Isomorphisms

The exceptional isomorphisms of alternating groups A_n with projective special linear groups \mathrm{[PSL](/page/PSL)}(2,q) for small n represent non-standard identifications that arise in the classification of finite simple groups and have deep ties to geometry and number theory. These isomorphisms were first established in the 1870s by Camille Jordan and Felix Klein, who linked the permutation actions of A_n to linear fractional transformations over finite fields, revealing structural parallels through shared orders and transitive actions on projective spaces.^[24] For n=4, the alternating group A_4 of order 12 is isomorphic to \mathrm{PSL}(2,3), the group of projective linear transformations over the field with 3 elements. This identification follows from the transitive action of \mathrm{PSL}(2,3) on the 4 points of the projective line over \mathbb{F}_3, mirroring the even permutations on 4 letters, with the isomorphism constructed via coset decompositions of stabilizers.^[24] The most prominent case is n=5, where A_5 of order $60is isomorphic to\mathrm{PSL}(2,5) \cong \mathrm{SL}(2,5)/{\pm I}, with \mathrm{SL}(2,5)denoting the special linear group over\mathbb{F}_5and{\pm I}its center. Jordan and Klein derived this by showing that the icosahedral rotation group, isomorphic toA_5, acts equivalently to \mathrm{PSL}(2,5)on the 6 points at infinity in the projective plane, via the Klein correspondence that equates quadratic forms to lines in projective geometry. The explicit isomorphism can be realized through the natural action of\mathrm{PSL}(2,5)on the cosets of a Borel subgroup, yielding a primitive permutation representation of degree 6 identical to that ofA_5$.^[24] For n=6, A_6 of order $360is isomorphic to\mathrm{PSL}(2,9), established through an outer [automorphism](/page/Automorphism) of the double cover 2 \cdot A_6 \cong \mathrm{SL}(2,9), where the isomorphism descends to the quotients by mapping even [permutations](/page/Permutation) to projective transformations over \mathbb{F}_9. This connection, also uncovered by Klein in his studies of modular equations, relies on the transitive action of \mathrm{PSL}(2,9)on 10 points (the [projective line](/page/Projective_line) over\mathbb{F}_9), aligning with A_6$'s permutation degree via geometric realizations in the complex plane.^[24]

A₄ and S₄

The alternating group A_4 is the subgroup of the symmetric group S_4 consisting of all even permutations of four elements, and thus has order |A_4| = 12.^[25] As a concrete realization within S_4, A_4 contains all 3-cycles and all products of two disjoint transpositions.^[25] Geometrically, A_4 is isomorphic to the group of rotational symmetries of a regular tetrahedron, which also has order 12, corresponding to the 12 possible orientations obtained by rotating the tetrahedron around its vertices, edges, or faces.^[26] A defining feature of A_4's structure as a subgroup of S_4 is its unique normal Sylow 2-subgroup, known as the Klein four-group V = \{e, (1\,2)(3\,4), (1\,3)(2\,4), (1\,4)(2\,3)\}, which consists of the identity and the three double transpositions in A_4.^[25] This subgroup V is normal in A_4 because conjugation by any element of A_4 permutes the double transpositions among themselves while preserving the even permutation structure.^[25] The Sylow structure of A_4 further highlights its organization: it has a single Sylow 2-subgroup of order 4 (namely V), and four Sylow 3-subgroups of order 3, each generated by a 3-cycle such as \langle (1\,2\,3) \rangle, \langle (1\,2\,4) \rangle, \langle (1\,3\,4) \rangle, and \langle (2\,3\,4) \rangle.^[25] The quotient A_4 / V is isomorphic to \mathbb{Z}/3\mathbb{Z}, the cyclic group of order 3 (equivalently, A_3), reflecting the action of A_4 on the cosets of V.^[25] This normal subgroup V also serves as the derived subgroup A_4', the subgroup generated by all commutators in A_4, which implies that A_4 is not simple and has abelianization A_4 / A_4' \cong \mathbb{Z}/3\mathbb{Z}.^[25] One explicit presentation of A_4 arises from its generation by the 3-cycles (1\,2\,3) and (1\,2\,4), with relations that enforce the normality of V:

A_4 \cong \langle x, y \mid x^2 = y^3 = (xy)^3 = 1 \rangle,

where x can be taken as a double transposition like (1\,2)(3\,4) and y as a 3-cycle like (1\,2\,3), ensuring the structure collapses appropriately to yield the Klein four-group as the commutator subgroup.^[27] This presentation underscores A_4's position as a semidirect product V \rtimes \mathbb{Z}/3\mathbb{Z} within S_4.^[25]

A₅ and 3D Rotations

The alternating group A_5 is isomorphic to the group of proper rotations (orientation-preserving symmetries) of the regular icosahedron and its dual, the regular dodecahedron, which embeds naturally into the special orthogonal group SO(3). This rotational icosahedral group has order 60, consistent with |A_5| = 5!/2 = 60.^[28]^[29] The group is generated by rotations of specific angles about axes passing through pairs of opposite vertices, faces, or edges of the icosahedron. Specifically, there are rotations by $72^\circ, $144^\circ, $216^\circ, and $288^\circ (order 5) about axes through opposite vertices (12 axes, contributing 24 non-identity rotations); rotations by $120^\circ and $240^\circ (order 3) about axes through centers of opposite faces (10 axes, contributing 20 non-identity rotations); and rotations by $180^\circ (order 2) about axes through midpoints of opposite edges (15 axes, contributing 15 rotations), plus the identity. These elements satisfy the relations of A_5, providing a geometric realization of its structure.^[28]^[30] The binary icosahedral group, a central extension of A_5 by \mathbb{Z}/2\mathbb{Z}, serves as its universal double cover and is isomorphic to the special linear group SL(2,5) of order 120. This double cover arises in the context of spin representations and the universal covering group of SO(3), where SL(2,5) \to A_5 is the quotient by the center \{ \pm I \}.^[31]^[32] As the unique non-abelian simple group of order less than 100, A_5 plays a foundational role in the classification of finite simple groups, with its icosahedral realization highlighting connections between discrete geometry and abstract algebra. The action of A_5 on the 12 vertices of the icosahedron yields a transitive permutation representation of degree 12, faithful and primitive, embedding A_5 into S_{12}.^[33]^[34]

The 15 Puzzle

The 15 puzzle is a classic sliding puzzle played on a 4×4 grid, featuring 15 square tiles numbered from 1 to 15 and one empty space, known as the blank. The goal is to rearrange the tiles into ascending order, row by row from left to right and top to bottom, by repeatedly sliding an adjacent tile into the blank space. Each move effectively transposes the blank with a neighboring tile, altering the positions of the pieces within the grid.^[35] The solvability of a 15 puzzle configuration depends on the parity of the permutation. Treating the blank as tile 16, any configuration corresponds to a permutation σ in the symmetric group S_{16}. The puzzle is solvable if and only if σ is an even permutation, i.e., the number of inversions in σ is even. An inversion is a pair of positions (i, j) with i < j but σ(i) > σ(j) when reading the grid row by row. The sign of the permutation is given by \operatorname{sgn}(\sigma) = (-1)^{\text{number of inversions}}, and solvability requires sgn(σ) = 1. This condition ensures the configuration lies within the reachable component of the puzzle's configuration space, which has size 16!/2 and corresponds to the even permutations in S_{16}. The arrangements of the 15 tiles with the blank in its target position form a set isomorphic to A_{15}.^[35]^[36] This parity invariant highlights the role of the alternating group in restricting reachable states: odd permutations, such as swapping tiles 14 and 15 while leaving others fixed (with blank in place), cannot be achieved. For the generalized n × n sliding puzzle with n² - 1 tiles, solvability requires an even permutation of all n² positions (including blank). When n is odd, this is equivalent to an even permutation of the tiles alone (i.e., the number of inversions among the tiles is even), as the blank's position parity aligns automatically. When n is even, as in the 15 puzzle, the condition accounts for the blank's position, but exactly half of all possible configurations are solvable.^[37]^[35]

Subgroups and Homology

Normal Subgroups and Simplicity

The alternating group A_3 is isomorphic to the cyclic group \mathbb{Z}/3\mathbb{Z}, which is simple as it has prime order and thus only trivial normal subgroups.https://kconrad.math.uconn.edu/blurbs/grouptheory/Ansimple.pdf In contrast, A_4 is not simple, as it contains the Klein four-group V = \{ e, (12)(34), (13)(24), (14)(23) \} as a nontrivial proper normal subgroup of order 4.http://ramanujan.math.trinity.edu/rdaileda/teach/s19/m3362/alternating.pdf For n \geq 5, the alternating group A_n is simple, meaning its only normal subgroups are the trivial subgroup \{e\} and A_n itself.https://kconrad.math.uconn.edu/blurbs/grouptheory/Ansimple.pdf This theorem establishes A_n (with n \geq 5) as non-abelian simple groups, forming one of the infinite families in the classification of finite simple groups alongside cyclic groups of prime order, groups of Lie type, and sporadic groups.https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-14/issue-1/Classifying-the-finite-simple-groups/bams/1183552783.pdf To sketch the proof, first note that all 3-cycles in A_n are conjugate for n \geq 3, and A_n is generated by its 3-cycles for n \geq 3.

\] Suppose $ N \trianglelefteq A_n $ is a nontrivial [normal subgroup](/page/Normal_subgroup). Then $ N $ must contain some even [permutation](/page/Permutation); analysis of possible [cycle](/page/Cycle) types shows that $ N $ contains a 3-cycle, as other even permutations (like double transpositions for $ n \geq 5 $) lead to contradictions via centralizer or index arguments.\[](https://www.math.purdue.edu/~jlipman/503/A_n.pdf

By conjugacy, N then contains all 3-cycles, and thus N = A_n since the 3-cycles generate A_n.

\) More precisely, if $ N \trianglelefteq A_n $ is [normal](/page/Normal), then either $ N $ contains all 3-cycles (hence $ N = A_n $) or $ N = \{ e \} $.\[https://kconrad.math.uconn.edu/blurbs/grouptheory/Ansimple.pdf

The base case n=5 follows by direct computation: any proper normal N would have order dividing 20 but cannot contain 3-cycles, 5-cycles, or double transpositions without being the full group.https://www.math.purdue.edu/~jlipman/503/A_n.pdf For n > 5, induction applies using the simplicity of A_{n-1} and examining N \cap A_{n-1}, showing nontrivial elements in N force it to be the full A_n.$$]

Other Subgroups

The point stabilizer of A_n in its natural transitive action on the set \{1, 2, \dots, n\} is isomorphic to A_{n-1}, consisting of all even permutations that fix a given point, and it has index n in A_n.^[22] This subgroup is maximal for n \geq 5. More generally, the stabilizer of a k-element subset, for $1 \leq k < n/2, is isomorphic to (S_k \times S_{n-k}) \cap A_n, which consists of pairs of permutations (\sigma, \tau) with \operatorname{sgn}(\sigma) \operatorname{sgn}(\tau) = 1, and has index \binom{n}{k}.^[22] These intransitive stabilizers form a family of maximal subgroups when appropriately chosen. For n \geq 5, the maximal subgroups of A_n also include imprimitive examples arising from equitable partitions of the n points into m blocks of equal size k (with n = km and m \geq 2), given by the even permutations in the wreath product S_k \wr S_m.^[22] The index of such a subgroup is n! / (k!^m m!). These imprimitive maximal subgroups complement the intransitive ones in the classification provided by the O'Nan–Scott theorem adapted to alternating groups. Primitive maximal subgroups exist as well, including those of affine, diagonal, and almost simple types, but their explicit structures depend on the value of n.^[22] The Sylow p-subgroups of A_n, for primes p dividing |A_n| = n!/2, have order equal to the highest power of p dividing n!/2, and their structure varies with p and n; for instance, the Sylow 2-subgroups are often described recursively via iterated wreath products of cyclic groups of order 2, reflecting the binary tree-like decomposition of even permutations.^[38] Subgroups isomorphic to smaller alternating groups A_k for k < n embed naturally into A_n by restricting to even permutations supported on a fixed k-element subset while fixing the remaining points pointwise; such an embedding has index n!/k!.^[39] These embeddings are not maximal in general but illustrate the hierarchical subgroup structure of A_n.

Abelianization (H₁)

In group homology, the first homology group H_1(G, \mathbb{Z}) of a group G with coefficients in the integers \mathbb{Z} is isomorphic to the abelianization G^{\mathrm{ab}} = G / [G, G], where [G, G] denotes the derived subgroup (commutator subgroup) generated by all commutators [g, h] = g h g^{-1} h^{-1} for g, h \in G.^[40] For the alternating group A_n, the abelianization depends on n. When n \leq 3, A_n is abelian (A_1 and A_2 are trivial, while A_3 \cong \mathbb{Z}/3\mathbb{Z}), so [A_n, A_n] = \{e\} and H_1(A_n, \mathbb{Z}) \cong A_n. For n = 4, the derived subgroup [A_4, A_4] is the Klein four-group V_4 = \{e, (12)(34), (13)(24), (14)(23)\}, which has order 4, yielding |A_4 : [A_4, A_4]| = 3 and H_1(A_4, \mathbb{Z}) \cong \mathbb{Z}/3\mathbb{Z}.^[12] For n \geq 5, A_n is perfect, meaning [A_n, A_n] = A_n. This follows from the fact that A_n is generated by 3-cycles and every 3-cycle in A_n can be expressed as a commutator of elements in A_n; specifically, a 3-cycle (abc) equals [(ab)(de), (acd)] for distinct d, e \notin \{a,b,c\}, ensuring the commutator lies in A_n. Thus, the derived subgroup, generated by such commutators of 3-cycles, equals A_n, and [ A_n / [A_n, A_n] = {e},

so $H_1(A_n, \mathbb{Z}) = \{0\}$.[](https://www-users.cse.umn.edu/~garrett/m/algebra/notes_2023-24/alternating_groups)[](http://ramanujan.math.trinity.edu/rdaileda/teach/s19/m3362/alternating.pdf) This trivial abelianization for $n \geq 5$ implies that $A_n$ admits no nontrivial abelian quotients, reflecting its non-abelian [simple](/page/Simple) structure with no abelian factors in its [composition series](/page/Composition_series).[](https://www-users.cse.umn.edu/~garrett/m/algebra/notes_2023-24/alternating_groups) ### Schur Multiplier (H₂) The Schur multiplier of a group $G$, denoted $M(G)$, is defined as the second [homology](/page/Homology) group $H_2(G, \mathbb{Z})$ with integer coefficients. It is isomorphic to the [kernel](/page/Kernel) of the [canonical](/page/Canonical) surjection from the universal central extension of $G$ onto $G$.[](https://arxiv.org/pdf/1812.04704) For the alternating group $A_n$ with $n \geq 4$, the Schur multiplier $M(A_n)$ is cyclic of order 2 when $n=4,5$ or $n \geq 8$, and cyclic of order 6 when $n=6$ or $n=7$.\[](https://groupprops.subwiki.org/wiki/Projective_representation_theory_of_alternating_group:A4)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A5)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A6)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A7)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A8) The Schur multiplier can be computed using the Hopf formula: given a free presentation $G \cong F/R$ where $F$ is a free group, then $H_2(G, \mathbb{Z}) \cong (R \cap [F, F]) / [R, F]$.[](https://arxiv.org/pdf/1812.04704) This approach has been applied to presentations of $A_n$, confirming the values above through explicit calculations of the intersection and commutator subgroups in the free group generated by transpositions or 3-cycles.[](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-10/issue-2/Review--F-Rudolf-Beyl-and-J%25C3%25BCrgen-Tappe-Group-extensions/bams/1183551591.pdf) These non-trivial multipliers correspond to stem extensions, which are central extensions where the [kernel](/page/Kernel) is contained in both the derived [subgroup](/page/Subgroup) and [the center](/page/The_Center) of the covering group. For $n=4,5$ and $n \geq 8$, the universal central extension is the double cover $2.A_n$ with [center](/page/The_Center) $\mathbb{Z}/2\mathbb{Z}$. For $n=6$ and $n=7$, it is the 6-fold cover $6.A_n$ with [center](/page/The_Center) $\mathbb{Z}/6\mathbb{Z}$. For $n=4$, the double cover is the [special linear group](/page/Special_linear_group) $ \mathrm{SL}(2,3) $.\[](https://groupprops.subwiki.org/wiki/Special_linear_group:SL(2,3))[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A5)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A6)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A7)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A8) In summary, $|M(A_n)| = 2$ for $n=4,5$ and $n \geq 8$, and $|M(A_n)| = 6$ for $n=6,7$.\[](https://groupprops.subwiki.org/wiki/Projective_representation_theory_of_alternating_group:A4)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A5)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A6)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A7)[](http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A8)

References

[1]
[PDF] The Alternating Group
The alternating group An is defined to be the kernel of ϵ: An = kerϵ. This means that An consists of all even permutations and is normal in Sn. Since Lemma ...Missing: A_n | Show results with:A_n
[2]
None
### Definition and Properties of Alternating Group A_n
[3]
[PDF] homomorphisms.pdf - Keith Conrad
The sign homomorphism sgn: Sn → {±1} has kernel An, the alternating group on n letters. Example 7.7. The homomorphism Aff(R) → R× from Example 3.3, where (a b.
[4]
[PDF] DETERMINANTS 1. Symmetric group 1.1. Groups. Let G be a ... - Math
An even permutation is a product of even number of transpositions. Page 5. DETERMINANTS. 5. 2. Determinants. 2.1. Definition of the determinant. Let K be a ...
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[PDF] Lecture 3: Homomorphisms and Isomorphisms
The kernel defines a new group, called the alternating group An := ker(sign). −1. 16For inverse, consider y ∈ im(f ). Then there exists x such that y = f(x) ...
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[PDF] MAS 305 Algebraic Structures II
The kernel of sign is a normal subgroup of Sn, called the alternating group An. It consists of all the even permutations. By a preceding theorem, the number of ...<|control11|><|separator|>
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[PDF] Supplement. The Alternating Groups A are Simple for n ≥ 5
Jan 10, 2013 · Alternating groups An are of order n!/2 and are only defined for n ≥ 2. When n = 2, A2 = {e} is the trivial group and so is not simple.
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Earliest Known Uses of Some of the Words of Mathematics (A)
ALTERNATING GROUP appears in Camille Jordan, "Sur la limite de transitivité des groupes non alternés," Bull. Soc. math. de France 1 (1873). Alternating group ...
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[PDF] Alternating Groups: - BillCookMath.com
Notice A1 really is not even defined (although some might declare this to be the trivial group), A2 is the trivial group, and A3 = ⟨(123)⟩ is cyclic. For ...
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[PDF] Sign of permutations - Keith Conrad
The identity permutation is even, and by Theorem 2.10 the product of even permutations is even. A permutation and its inverse are a product of the same number ...
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[PDF] An is generated by the 3-cycles
Mar 5, 2008 · The idea of the proof is that we will show that any product of two trans- positions is a product of 3-cycles; and, if so, then since every ...
[12]
[PDF] The Alternating Group
(4) An is called the alternating group. An important feature of the alternating group is that, unless n = 4, it is a simple group. A group G is said to be ...
[13]
[PDF] Lecture #14 of 24 ∼ October 19th, 2020 - Math 5111 (Algebra 1)
Oct 19, 2020 · The alternating group An is the kernel of the sign map and is therefore is a normal subgroup of Sn. Explicitly, An consists of all even ...
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https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/04%3A_Families_of_Groups/4.04%3A_Alternating_Groups
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alternating group is a normal subgroup of the symmetric group
Mar 22, 2013 · alternating group is a normal subgroup of the symmetric group ... index 2 2 in Sn S n . In general, if a subgroup ...
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Alternating Group -- from Wolfram MathWorld
An alternating group is a group of even permutations on a set of length n, denoted A_n or Alt(n), and is a permutation group.
[17]
[PDF] 21 Conjugacy Classes for Symmetric and Alternating Groups
These examples demonstrate that after our analysis of cycle types in symmetric and alternating groups, deter- mining the class equation is not so much algebra ...
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Splitting criterion for conjugacy classes in the alternating group
May 30, 2012 · Cycle type determines conjugacy class · symmetric group on finite set, also symmetric group on infinite set, The conjugacy class of an element ...1Statement · 2Related facts · 2.1Related facts for symmetric...
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[PDF] SIMPLICITY OF An - KEITH CONRAD
The group A3 is simple, since it has order 3, and the groups A1 and A2 are trivial. We will give five proofs of Theorem 1.1. Section 2 includes some ...Missing: A_1 A_2
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[PDF] Short Presentations for alternating and symmetric groups
For every integer n > 2, the alternating group An has: (a) a 2-generator presentation with O(log n) relations and length O(log2 n); (b) a presentation of ...
[21]
[PDF] What is a group presentation? - Cornell Mathematics
Consider all possible 2-generator groups G = (a, b) in which the generators satisfy the 3 relations a3 = b3 = (ab)2 = 1. For example:.Missing: A_n | Show results with:A_n
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[PDF] Chapter 1 The alternating groups
Numbering the cosets of H from 1 to n, where n = |G : H|, we obtain a permu- tation action of G on these n points, or in other words a group homomorphism from G ...
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[PDF] arXiv:math/0311462v1 [math.AG] 26 Nov 2003
The alternating group A6 plays a very special role in the theory of finite groups ... outer automorphism group Out(A6) is isomorphic to a bigger group C⊕2. 2 ...Missing: A_6 | Show results with:A_6
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[PDF] The Geometry of the Classical Groups
is isomorphic to A5, a simple group of order 168 is isomorphic to. PSL(3, 2) and that A6 is isomorphic to PSL(2, 9). 6.6. Let p be a prime, set q = pr and ...
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[PDF] Abstract Algebra
... Dummit. University of Vermont. Richard M. Foote. University of Vermont john ... algebraic topology, algebraic geometry, representation theory, Lie groups, etc.
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[PDF] Math 120A — Introduction to Group Theory - UCI Mathematics
Oct 29, 2025 · Consider the alternating group An as a subgroup of Sn. Generalizing the argument from Theo- rem 4.25, we see that for any α ∈ An and σ ∈ Sn,.Missing: A_n | Show results with:A_n
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[PDF] Math 350 Introduction to Abstract Algebra Fall 2017 Extra Problems
Prove that an integer n > 1 is a prime number if and only if (n − 1)! ... (a) A4 = hx, y | x2 = y3 = (xy)3 = 1i. (b) S4 = hx, y | x2 = y3 = (xy)4 = 1i.
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[PDF] Lecture 20: Class Equation for the Icosahedral Group
Theorem 20.6. The icosahedral group I is isomorphic to the alternating group A5. Proof. We want to show that an element in I acts in the same way as an ...
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[PDF] ICOSAHEDRON SYMMETRY GROUP AND ROTATION MATRICES
... rotational symmetry group of icosahedron is isomorphic to A5 (A stands for alternating group) A5 → SO3(R). Representation theory. A representation of a group ...
[30]
[PDF] The Irreducible Representations of A5
The icosahedron, the 3-dimensional platonic solid consisting of 20 equilateral triangles, has symmetry group related to A5. This fact can be used to find a.
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[PDF] Notes on Classical Groups
Remark: This construction explains why the groups SL(2,3) and SL(2,5) are sometimes called the binary tetrahedral and binary icosahedral groups. Construct.
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[PDF] Finite subgroups of SL(2, F) and automorphy
SL(2,5) has a normal subgroup Z = h−Ii of order 2, and the quotient SL(2,5)/Z is isomorphic to A5. SL(2,5) has no subgroup isomorphic to A5. Of course, S5 does.
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[PDF] the sylow theorems and their applications - UChicago Math
The smallest non-abelian simple group is A5, and is unique up to isomorphism. Proof. In order to prove this theorem, we first eliminate all possibilities of ...
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[PDF] arXiv:1404.3170v2 [math.DS] 1 Jan 2017
Jan 1, 2017 · Indeed, it coincides with the icosahedron's face-centers. There are two other special icosahedral orbits: 12 vertices and 30 edge- midpoints.
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[PDF] the 15-puzzle (and rubik's cube) - Keith Conrad
In summary, half the permutations of the pieces 1 to 15 in the 15-puzzle are solvable since A15 is generated by 3-cycles with a common pair of terms, like 11 ...Missing: alternating | Show results with:alternating
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[PDF] CRACKING THE 15 PUZZLE - UCLA Math Circle
Theorem 2. Configurations of the 15 puzzle have “similar parity” when the number of inversions and the taxicab distance from the blank space to the lower right ...
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[PDF] Permutations and the 15-Puzzle
Jan 21, 2004 · The set of even permutations is called the alternating group on n letters and denoted An. Exercises. 1. Write down the three elements of A3 and ...
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[PDF] Homework 1: Solving the Sliding Puzzle With A*
3. If the puzzle's width is odd, the puzzle is solvable if and only if the total number of inversions is even. 4. If the puzzle's width is even, you must add ...
[39]
Structure and minimal generating sets of Sylow 2-subgroups ... - arXiv
Feb 19, 2017 · The purpose of this paper is to research the structure of a Sylow 2-subgroups and to construct a minimal generating system for such subgroups.
[40]
Permutation Groups | SpringerLink
This text can serve as an introduction to permutation groups in a course at the graduate or advanced undergraduate level, or for self- study.
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[PDF] Altenating grouops, commutator subgroups
Jan 22, 2024 · The commutator subgroup [G, G] of G is the smallest subgroup of G containing all commutators. [2] It is immediate that G is abelian if and ...
[42]
[PDF] Schur Multipliers and Second Quandle Homology - arXiv
Dec 11, 2018 · The book by Beyl and Tappa [BT] gives the definition of the Schur multiplicator, commonly known as the Schur multiplier, M(G) via the Schur ...Missing: A_n | Show results with:A_n
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http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A5
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http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A6
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http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A7
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http://brauer.maths.qmul.ac.uk/Atlas/v3/group/A8
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Group extensions, representations, and the Schur multiplicator, by F ...
Schur multipliers arise when one studies central extensions of groups. A central extension is a surjective homomorphism <p: G -> Q whose kernel is contained in ...