In mathematics, an alternating group is the group of even permutations of a finite set. 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 An or Alt(n).
The group A4 has the Klein four-group V as a proper normal subgroup, namely the identity and the double transpositions { (), (12)(34), (13)(24), (14)(23) }, that is the kernel of the surjection of A4 onto A3 ≅ Z3. We have the exact sequenceV → A4 → A3 = Z3. In Galois theory, this map, or rather the corresponding map S4 → S3, corresponds to associating the Lagrange resolvent 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, any two elements of An that are conjugate by an element of An 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 (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.
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, An 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 An is simple for n ≥ 5.
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-groupV = Z2 × Z2, 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(123)(456)).
A5 is isomorphic to PSL2(4), PSL2(5), and the symmetry group of chiral icosahedral symmetry. (See[1] for an indirect isomorphism of PSL2(F5) → A5 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 SL1(q) = PSL1(q) for any q).
Cayley table of the alternating group A4 Elements: The even permutations (the identity, eight 3-cycles and three double-transpositions (double transpositions in boldface))
icosahedron – radius 4π/5 – half of the split conjugacy class of 5-cycles
dodecahedron – radius 2π/3 – conjugacy class of 3-cycles
icosahedron – radius 2π/5 – second half of split 5-cycles
Compound of five tetrahedra. A5 acts on the dodecahedron by permuting the 5 inscribed tetrahedra. Even permutations of these tetrahedra are exactly the symmetric rotations of the dodecahedron and characterizes the A5 < SO3(R) correspondence.
A5 is the group of isometries of a dodecahedron in 3-space, so there is a representation A5 → SO3(R).
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 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 π radians, and so can be represented by a vector of length π 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 2π/5 and 4π/5, respectively. The nontrivial outer automorphism in Out(A5) ≃ Z2 interchanges these two classes and the corresponding icosahedra.
It can be proved that the 15 puzzle, a famous example of the sliding puzzle, can be represented by the alternating group A15,[2] because the combinations of the 15 puzzle can be generated by 3-cycles. In fact, any 2k − 1 sliding puzzle with square tiles of equal size can be represented by A2k−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 |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 distinct nontrivial element generates the whole group.
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).
This is easily seen directly, as follows. An is generated by 3-cycles – so the only non-trivial abelianization maps are An → Z3, since order-3 elements must map to order-3 elements – and for n ≥ 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, An 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 A3 ↠ Z3 (in fact an isomorphism) and A4 ↠ Z3.
The Schur multipliers 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 also a triple cover. In these cases, then, the Schur multiplier is (the cyclic group) of order 6.[3] These were first computed in (Schur 1911).
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References
Robinson, Derek John Scott (1996), A course in the theory of groups, Graduate texts in mathematics, vol.80 (2ed.), Springer, ISBN978-0-387-94461-6
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