The

Alternating group *A*_{n} is the

group of
all even

permutations of

*{1,2,...,n}*. It is a

normal subgroup
of

*S*_{n}, the

Symmetric group.

*A*_{n} has *n! /2* elements.

For *n>=5*, *A*_{n} is a simple
group. This is the essential reason why
quintic polynomials are not soluble by radicals.