stabilizer

stabilizes

stabilized

is.stabilizer

A permutation \(\phi\) is said to <dfn>stabilize</dfn> 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].

Manipulates invertible functions from a finite set to
itself. Can transform from word form to cycle form and
back. To cite the package in publications please use
Hankin (2020) "Introducing the permutations R package",
SoftwareX, volume 11 <doi:10.1016/j.softx.2020.100453>.

Robin K S Hankin

permutations

The Symmetric Group: Permutations of a Finite Set

Paul Egeler

stabilizer function

<dl> <dt>a</dt>
<dd>Permutation (coerced to class <code>cycle</code>)</dd> <dt>s</dt>
<dd>Subset of \(\left\lbrace
 1,\ldots,n\right\rbrace\), to be stabilized</dd></dl>

Arguments

Author

A permutation \(\phi\) is said to <dfn>stabilize</dfn> 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].

Stabilizer of a permutation — stabilizer

<dl>

 <dt>a</dt>
<dd>Permutation (coerced to class <code>cycle</code>)</dd>

 <dt>s</dt>
<dd>Subset of \(\left\lbrace
 1,\ldots,n\right\rbrace\), to be stabilized</dd>

</dl>

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stabilizer: Stabilizer of a permutation

Description

Usage

Value

Arguments

Author

Examples