View source for Alternating group

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''of degree $n$''

The subgroup $A_n$ of the [[Symmetric group|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 $S_n$ 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.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Hall,   "Group theory" , Macmillan  (1959)</TD></TR></table>



====Comments====
Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012030/a01203019.png" /> is the non-Abelian simple group of smallest possible order.