permutations (version 1.1-6)

stabilizer: Stabilizer of a permutation

Description

A permutation \(\phi\) is said to stabilize a set \(S\) if the image of \(S\) under \(\phi\) is a subset of \(S\), that is, if \(\left\lbrace\left. \phi(s)\right|s\in S \right\rbrace\subseteq S \). This may be written \(\phi(S)\subseteq S\). Given a vector \(G\) of permutations, we define the stabilizer of \(S\) in \(G\) to be those elements of \(G\) that stabilize \(S\).

Given \(S\), it is clear that the identity permutation stabilizes \(S\), and if \(\phi,\psi\) stabilize \(S\) then so does \(\phi\psi\), and so does \(\phi^{-1}\) [\(\phi\) is a bijection from \(S\) to itself].

Usage

stabilizes(a,s)
stabilizer(a,s)

Value

A boolean vector [stabilizes()] or a vector of permutations in cycle form [stabilizer()]

Arguments

a

Permutation (coerced to class cycle)

s

Subset of \(\left\lbrace 1,\ldots,n\right\rbrace\), to be stabilized

Author

Robin K. S. Hankin

Examples

a <- rperm(200)
stabilizer(a,3:4)

all_perms_shape(c(1,1,2,2)) |> stabilizer(2:3)  # some include (23), some don't