C5 - transposition

transposition

a $2$ -cycle is called a transposition

a $3$ -cycle can be broken into two $2$ -cycles:

(\begin{matrix} a & b & c \end{matrix}) = (\begin{matrix} a & b \end{matrix}) (\begin{matrix} a & c \end{matrix})

the number of $2$ -cycles define whether a permutation is even or odd
the above example is an even permutation as it as an even number of $2$ -cycles
$A_{n}$ is the group of all even permutations of $S_{n}$
$S_{n} \ A_{n}$ , ie. the odd permutations, do not form a group, a composition may lead to even permutations, so, not closed

permutations are either even or odd

$S_{n} = A_{n} \cup (S_{n} \ A_{n})$

the order of $A_{n}$ is $n! / 2$
there are equal number of elements in even and odd permutations as a bijection can be formed between them
consider $α \in S_{8}$ which is formed of a $3$ - and a $4$ -cycle, so $| α | = 12$
${α, α^{2}, α^{3}, \dots α^{11}} = ⟨ α ⟩ \leq S_{8}$ , ie. it is a subgroup of $S_{8}$
therefore, $| α | \leq | S_{8} |$