Alternating group
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Algebraic structure → Group theory Group theory |
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In
Basic properties
For n > 1, the group An is the commutator subgroup of the symmetric group Sn with index 2 and has therefore n!/2 elements. It is the kernel of the signature group homomorphism sgn : Sn → {1, −1} explained under symmetric group.
The group An is
The group A4 has the
Conjugacy classes
As in the
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.
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.
Automorphism group
n | Aut(An) | Out(An) |
---|---|---|
n ≥ 4, n ≠ 6 | Sn | Z2 |
n = 1, 2 | Z1 | Z1 |
n = 3 | Z2 | Z2 |
n = 6 | S6 ⋊ Z2 | V = Z2 × Z2 |
For n > 3, except for n = 6, the
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. 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
- A4 is isomorphic to PSL2(3)[1] 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[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).
Examples S4 and A4
A3 = Z3 (order 3) |
A4 (order 12) |
A4 × Z2 (order 24) |
S3 = Dih3 (order 6) |
S4 (order 24) |
A4 in S4 on the left |
Example A5 as a subgroup of 3-space rotations
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.
Example: the 15 puzzle
It can be proved that the
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.
For all n > 4, An has no nontrivial (that is, proper) normal subgroups. Thus, An is a simple group for all n > 4. A5 is the smallest non-solvable group.
Group homology
The
H1: Abelianization
The first
- H1(An, Z) = Z1 for n = 0, 1, 2;
- H1(A3, Z) = Aab
3 = A3 = Z3; - H1(A4, Z) = Aab
4 = Z3; - H1(An, Z) = Z1 for n ≥ 5.
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.
H2: Schur multipliers
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).
- H2(An, Z) = Z1 for n = 1, 2, 3;
- H2(An, Z) = Z2 for n = 4, 5;
- H2(An, Z) = Z6 for n = 6, 7;
- H2(An, Z) = Z2 for n ≥ 8.
Notes
- ^ a b Robinson (1996), p. 78
- ^ Beeler, Robert. "The Fifteen Puzzle: A Motivating Example for the Alternating Group" (PDF). faculty.etsu.edu/. East Tennessee State University. Archived from the original (PDF) on 2021-01-07. Retrieved 2020-12-26.
- ^ Wilson, Robert (October 31, 2006), "Chapter 2: Alternating groups", The finite simple groups, 2006 versions, archived from the original on May 22, 2011, 2.7: Covering groups
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References
- Robinson, Derek John Scott (1996), A course in the theory of groups, Graduate texts in mathematics, vol. 80 (2 ed.), Springer, ISBN 978-0-387-94461-6
- S2CID 122809608
- Scott, W.R. (1987), Group Theory, New York: ISBN 978-0-486-65377-8