An alternating group is a type of mathematical group that consists of all the even permutations of a finite set. A permutation is a rearrangement of elements, and an even permutation is one that can be achieved by an even number of transpositions, which are simple swaps of two elements. The alternating group is denoted as A_n, where n represents the number of elements in the set. Alternating groups are important in the field of group theory, a branch of abstract algebra. They play a crucial role in understanding the symmetry of objects and are used in various areas of mathematics, including combinatorics and geometry. The smallest non-trivial alternating group is A_3, which has three elements and consists of the even permutations of three objects.