Cardinality of Even and Odd Permutations on Finite Set

Theorem

Let $n \in \N_{> 0}$ be a natural number greater than $0$.

Let $S$ be a set of cardinality $n$.

Let $S_n$ denote the symmetric group on $S$ of order $n$.

Let $R_e$ and $R_o$ denote the subsets of $S_n$ consisting of even permutations and odd permutations respectively.

Then the cardinality of both $R_e$ and $R_o$ is $\dfrac {n!} 2$.

Proof

From Order of Symmetric Group:

$\order {S_n} = n!$

where:

$\order {S_n}$ denotes the order of $S_n$

$n!$ denotes the factorial of $n$.

By definition:

$\card {R_e} + \card {R_o} = \order {S_n}$

From Odd and Even Permutations of Set are Equivalent:

$\card {R_e} = \card {R_o}$

The result follows.

$\blacksquare$

Sources

• 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory