Symmetric Group - Low Degree Groups

Low Degree Groups

The low-degree symmetric groups have simpler and exceptional structure, and often must be treated separately.

Sym(0) and Sym(1)

The symmetric groups on the empty set and the singleton set are trivial, which corresponds to 0! = 1! = 1. In this case the alternating group agrees with the symmetric group, rather than being an index 2 subgroup, and the sign map is trivial. In the case of Sym(0), its only member is the Empty function.

Sym(2)

This group consists of exactly two elements: the identity and the permutation swapping the two points. It is a cyclic group and so abelian. In Galois theory, this corresponds to the fact that the quadratic formula gives a direct solution to the general quadratic polynomial after extracting only a single root. In invariant theory, the representation theory of the symmetric group on two points is quite simple and is seen as writing a function of two variables as a sum of its symmetric and anti-symmetric parts: Setting f_s(x,y) = f(x,y) + f(y,x), and f_a(x,y) = f(x,y) − f(y,x), one gets that 2·f = f_s + f_a. This process is known as symmetrization.

Sym(3)

This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle. Cycles of length two correspond to reflections, and cycles of length three are rotations. In Galois theory, the sign map from Sym(3) to Sym(2) corresponds to the resolving quadratic for a cubic polynomial, as discovered by Gerolamo Cardano, while the Alt(3) kernel corresponds to the use of the discrete Fourier transform of order 3 in the solution, in the form of Lagrange resolvents.

Sym(4)

The group S₄ is isomorphic to the group of proper rotations about opposite faces, opposite diagonals and opposite edges, 9, 8 and 6 permutations, of the cube. Beyond the group Alt(4), Sym(4) has a Klein four-group V as a proper normal subgroup, namely the even transpositions {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}, with quotient Sym(3). In Galois theory, this map corresponds to the resolving cubic to a quartic polynomial, which allows the quartic to be solved by radicals, as established by Lodovico Ferrari. The Klein group can be understood in terms of the Lagrange resolvents of the quartic. The map from Sym(4) to Sym(3) also yields a 2-dimensional irreducible representation, which is an irreducible representation of a symmetric group of degree n of dimension below n−1, which only occurs for n=4.

Sym(5)

Sym(5) is the first non-solvable symmetric group. Along with the special linear group SL(2, 5) and the icosahedral group Alt(5) × Sym(2), Sym(5) is one of the three non-solvable groups of order 120 up to isomorphism. Sym(5) is the Galois group of the general quintic equation, and the fact that Sym(5) is not a solvable group translates into the non-existence of a general formula to solve quintic polynomials by radicals. There is an exotic inclusion map Sym(5) → Sym(6) as a transitive subgroup; the obvious inclusion map Sym(n) → Sym(n+1) fixes a point and thus is not transitive. This yields the outer automorphism of Sym(6), discussed below, and corresponds to the resolvent sextic of a quintic.

Sym(6)

Sym(6), unlike other symmetric groups, has an outer automorphism. Using the language of Galois theory, this can also be understood in terms of Lagrange resolvents. The resolvent of a quintic is of degree 6—this corresponds to an exotic inclusion map Sym(5) → Sym(6) as a transitive subgroup (the obvious inclusion map Sym(n) → Sym(n+1) fixes a point and thus is not transitive) and, while this map does not make the general quintic solvable, it yields the exotic outer automorphism of Sym(6)—see automorphisms of the symmetric and alternating groups for details.

Note that while Alt(6) and Alt(7) have an exceptional Schur multiplier (a triple cover) and that these extend to triple covers of Sym(6) and Sym(7), these do not correspond to exceptional Schur multipliers of the symmetric group.