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Group of even permutations of a finite set

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 or Alt(n).

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 explained under symmetric group. The group An is abelian if and only if n ≤ 3 and simple if and only if 1=n = 3 or n ≥ 5. A5 is the smallest non abelian simple group, having order 60, and the smallest non solvable group. The group A4 has the Klein four group V as a proper normal subgroup, namely the identity and the double transpositions , that is the kernel of the surjection of A4 onto 1=A3 ≅ Z3. We have the exact sequence 1=V → 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.

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.

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.

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) 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 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.

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 , there does not necessarily exist a subgroup of G with order d: the group 1=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.

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).