Groups, Symmetric and Alternating Groups

Symmetric and Alternating Groups

The symmetric group on n letters, written S_n, is the group of all possible permutations on n letters. The order of S_n is n!.

The alternating group on n letters, written A_n, is the group of all even permutations on n letters. The order of A_n is n!/2. A_n is normal in S_n, in fact it is the kernel of the parity homomorphism.

The group A_n+1 defines the group of rotations of a generalized tetrahedron in n space, while S_n+1 defines the group of rotations and reflections. This can be seen by placing any vertex in position, then the next, then the next, and so on, and reflecting if the last two must be swapped.

The symmetric group is the group of rotations and reflections of a tetrahedron

For S_∞, parity, and hence A_∞, may be well defined if permutations transpose only a finite number of letters in a linearly ordered set. Once again A_∞ is the kernel of the parity homomorphism, and is normal in S_∞.