An alternating group is a cyclic group with order 2. A cyclic group is a group that has only one subgroup of order 1. The alternating group is the group of perm...
An alternating group is a cyclic group with order 2. A cyclic group is a group that has only one subgroup of order 1. The alternating group is the group of permutations of the set {1, 2, 3}, with the operation of permutation given by combining two permutations in the order they are written. For example, the permutation (1, 2, 3) and (3, 1, 2) are equivalent, since they both map the element 1 to the element 2, and the element 2 to the element 3.
The alternating group has a unique property: every element has exactly two orbiters. An orbitor is an element that is mapped to every other element by a permutation. In this case, the orbiters are 1 and 3, since every permutation maps these elements to every other element.
The alternating group is an important group in group theory, since it provides a concrete example of a non-abelian group