Stabilizer of a permutation

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].

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

Robin K. S. Hankin

Examples

a <- rperm(200)
stabilizer(a,3:4)
#>  [1] (16725)       (12765)       (12576)       (15276)       (12756)(34)  
#>  [6] (16275)       (16752)       (156)(27)(34) (15627)(34)   (176)(34)    
#> [11] (34)(567)     (257)         (1752)        (156)(34)     (12)(576)    

all_perms_shape(c(1,1,2,2)) |> stabilizer(2:3)  # some include (23), some don't
#> [1] (16)(23) (16)(45) (14)(23) (14)(56) (15)(23) (15)(46) (23)(56) (23)(46)
#> [9] (23)(45)