Alternating group

of degree $n$

The subgroup $A_n$ of the symmetric group $S_n$ consisting of all even permutations. $A_n$ is a normal subgroup in $S_n$ of index 2 and order $n!/2$. The permutations of $A_n$, considered as permutations of the indices of variables $x_1,\ldots,x_n$, leave the alternating polynomial $\prod(x_i-x_j)$ invariant, hence the term "alternating group". The group $A_m$ may also be defined for infinite cardinal numbers $m$, as the subgroup of $S_m$ consisting of all even permutations. If $n>3$, the group $A_n$ is $(n-2)$-fold transitive. For any $n$, finite or infinite, except $n=4$, this group is simple; this fact plays an important role in the theory of solvability of algebraic equations by radicals.

Comments

Note that $A_5$ is the non-Abelian simple group of smallest possible order.

References

• [1] M. Hall, "Group theory" , Macmillan (1959) Zbl 0084.02202