The fact that sgn is multiplicative implies that products and inverses
of even permutations are even. This
gives rise to the following definition.
Definition
By A
n we denote the set of even permutations in
S
n.
This set is closed with respect to taking products and inverse elements.
We call A
n the
alternating group on
n letters.
There are just as many even as odd permutations in Sn.
Theorem
For
n > 1, the alternating
group A
n contains precisely
n!/2 elements.