stabilizer: Stabilizer of a permutation

In permutations: The Symmetric Group: Permutations of a Finite Set

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)

Arguments

a	Permutation (coerced to class cycle)
s	Subset of \left\lbrace 1,\ldots,n\right\rbrace, to be stabilized

Value

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

Note

The identity permutation stabilizes any set.

Functions stabilizes() and stabilizer() coerce their arguments to cycle form.

Author(s)

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

permutations documentation built on April 3, 2025, 7:09 p.m.