E3 - application of cosets to permutation groups

stabiliser

let $G$ be a group of permutations of a set $S$ , then for each $i$ in $S$ , then the stabiliser of $i$ in $G$ is defined as:

{stab}_{G} (i) = {ϕ \in G ∣ ϕ (i) = i}

note: ${stab}_{G} (i) \leq G$

orbit

${orb}_{G} (i) = {ϕ (i) ∣ ϕ \in G}$

the set ${orb}_{G} (i)$ is a subset of $S$ , called the orbit of $i$ under $G$

eg: let $G$ be the following subgroup of $S_{8} :$ ${(1), (132) (465) (78), (132) (465), (123) (456) (78), (78)}$
then:

\begin{matrix} {orb}_{G} (1) = {1, 3, 2}, & {stab}_{G} (1) = {(1), (78)}, \\ {orb}_{G} (2) = {2, 1, 3}, & {stab}_{G} (2) = {(1), (78)}, \\ {orb}_{G} (4) = {4, 5, 6}, & {stab}_{G} (4) = {(1), (78)}, \\ {orb}_{G} (5) = {7, 8}, & {stab}_{G} (7) = {(1), (132) (465)}, \end{matrix}