C1 - permutation and symmetry

permutation of a set

let $A$ be a set, then the permutation of set $A$ is a bijective function such that $f : A \to A$

note: a bijective function is a one-to-one and onto function
a permutation group of a set $A$ is a set of permutations of $A$ that forms a group under function composition
$A$ is taken to be a finite set of the form ${1, 2, 3, \dots, n}$ , then its permutation is defined as $α (a_{i}) = a_{j}$ , where $a_{i}, a_{j} \in A$
eg: for a set ${1, 2, 3, 4}$ : $α (1) = 2$ , $α (2) = 3$ , $α (3) = 4$ and $α (4) = 1$
it can also be expressed as an array:

α = [\begin{matrix} 1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1 \end{matrix}]

composition of permutations is carried out from right to left, ie: $(σ γ) (a_{i}) = σ (γ (a_{i}))$

symmetry

sym (A) := {f : A \to A ∣ f is bijective}

where, $f$ is a permutation of $A$

symmetry group

the symmetry equipped with function compositions $(sym (A), \circ)$ forms a group