Alternating group ($A_n$)

The alternating group $A_n$ is the subgroup of the symmetric group $S_n$ consisting of even permutations, those expressible as an even number of transpositions. It is the kernel of the sign homomorphism $S_n\to \{1,-1\}$, so it is a normal subgroup and has order $n!/2$ for $n\ge 2$.

A three-cycle is even because $(abc)=(ac)(ab)$, and three-cycles generate $A_n$. For example, $A_3$ is a cyclic group of order three. For $n\ge 5$, $A_n$ is a nonabelian simple group, a central family in finite group theory and the obstruction behind the general unsolvability of polynomial equations by radicals.

An **alternating group**, denoted \( A_n \), is a specific type of group in the field of abstract algebra. It consists of all the even permutations of a finite set of \( n \) elements. To fully understand this concept, it's important to break down a few terms: 1. **Permutation**: A permutation of a set is a rearrangement of its elements.