Alternating group

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.

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.

The extra outer automorphism in A6 swaps the 3-cycles (like (123)) with elements of shape 32 (like (123)(456)).

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 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.

In fact, any 2k − 1 sliding puzzle with square tiles of equal size can be represented by A2k−1.

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.

In these cases, then, the Schur multiplier is (the cyclic group) of order 6.

Cayley table of the symmetric group S 4

The odd permutations are colored:
Transpositions in green and 4-cycles in orange
Cayley table of the alternating group A 4
Elements: The even permutations (the identity, eight 3-cycles and three double- transpositions (double transpositions in boldface))

Subgroups:
Klein four-group
Cyclic group Z3 Cyclic group Z3 Cyclic group Z3 Cyclic group Z3
A 5 < SO 3 ( R )
ball – radius π – principal homogeneous space of SO(3)
icosidodecahedron – radius π – conjugacy class of 2-2-cycles
icosahedron – radius 4 π /5 – half of the split conjugacy class of 5-cycles
dodecahedron – radius 2 π /3 – conjugacy class of 3-cycles
icosahedron – radius 2 π /5 – second half of split 5-cycles
Compound of five tetrahedra. A 5 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 A 5 < SO 3 ( R ) correspondence.
A 15 puzzle .