stabilizer: Stabilizer of a permutation

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

Stabilizer of a permutation

Description

Permutations that leave elements unchanged

Usage

stabilizes(a, s, strict=FALSE)
stabilizer(a, s, strict=FALSE)
doesnotmove(a, s)

Arguments

a	Permutation
s	Subset of \left\lbrace 1,\ldots,n\right\rbrace, to be stabilized
strict	Boolean, see details

Details

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 do \phi\psi and \psi\phi, and so does \phi^{-1} [\phi is a bijection from S to itself].

Function stabilizes(G,S) returns a Boolean vector V with V[i] being TRUE if G[i] stabilizes S and FALSE otherwise. Function stabilizer(G,S) returns G[stabilizes(G,S)].

Sometimes we are interested in whether each element of S maps to itself, that is, \forall s\in S, \phi(s)=s. This is a stronger requirement than stabilization, which allows the elements of S to be permuted. To check for this, use strict=TRUE, which calls doesnotmove().

Function stabilizes() coerces its argument to cycle form; doesnotmove() is generic.

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.

Note that if \phi stabilizes S, then the elements of S may be permuted by \phi; thus some elements of S[i]^stabilizer(G,S) [equivalently, as.function(stabilizer(G,S))(S[i])] may not be equal to S[i].

Author(s)

Robin K. S. Hankin

Examples

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

3^s   # all these in {3,4}

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


a <- rperm(300, moved=4)
table(stab=stabilizes(a,1:2), dnm=doesnotmove(a,1:2)) # note zero at top right

permutations documentation built on July 13, 2026, 5:07 p.m.