90 Combinatorics of Permutations, Second Edition
PROPOSITION 3.9
The set of all even permutations in S
n
forms a subgroup.
This subgroup is called the alternating group of degree n, and is denoted
by A
n
. The reader should prove at this point that A
n
has n!/2 elements if
n ≥ 2, then she should check her answer in Exercise 1. Like the symmetric
group, the alternating group has been vigorously investigated throughout the
last century. For instance, it is known that A
n
is a simple group if n ≥ 5. In
fact, among all finite simple groups, A
n
is the easiest to define, except for the
cyclic groups Z
p
,wherep is a prime. (See any introductory book on group
theory for the definition of a simple group or Z
p
.) It is also interesting that A
n
is by far larger than any other ...