Alternating Group

An alternating group is a group of even permutations on a set of length , denoted or Alt() (Scott 1987, p. 267). Alternating groups are therefore permutation groups.

The th alternating group is represented in the Wolfram Language as AlternatingGroup[n].

An alternating group is a normal subgroup of the permutation group, and has group order , the first few values of which for , 3, ... are 1, 3, 12, 60, 360, 2520, ... (OEIS A001710). The alternating group is -transitive.

Amazingly, the pure rotational subgroup of the icosahedral group is isomorphic to . The full icosahedral group is isomorphic to the group direct product , where is the cyclic group on two elements.

Alternating groups with are simple groups (Scott 1987, p. 295), i.e., their only normal subgroups are the trivial subgroup and the entire group .

The number of conjugacy classes in the alternating groups for , 3, ... are 1, 3, 4, 5, 7, 9, ... (OEIS A000702).

is the only nontrivial proper normal subgroup of .

The multiplication table for is illustrated above.

The cycle index (in variables , ..., ) for the alternating group is given by

(1)

(Harary 1994, p. 184), where the sum runs over the set of solution vectors to

(2)

The cycle indices for the first few are

			(3)
			(4)
			(5)
			(6)
			(7)

Wolfram Web Resources