C5 - transposition

transposition

a $2$ -cycle is called a transposition

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

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

check:

\begin{matrix} a \to b \\ b \to a \to c \\ c \to a \end{matrix}

lemma

if the identity, $ϵ = β_{1} β_{2} \dots β_{r}$ , where the $β$ 's are 2-cycles, then $r$ is even

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
the group of even permutations of $n$ symbols is denoted by $A_{n}$ , called the alternating group of degree $n$

permutations are either even or odd

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

even permutations form a group

the set of even permutations in $S_{n}$ forms a subgroup of $S_{n}$

order of

A_{n}

the order of $A_{n}$ is $n! / 2$ , for $n > 1$

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, $| α | \leq | S_{8} |$