# PermutationGroup

PermutationGroup[{perm1,,permn}]

represents the group generated by multiplication of the permutations perm1,,permn.

# Details

• The generating permutations permi must be given in disjoint cyclic form, with head Cycles.
• Properties of a permutation group are typically computed by constructing a strong generating set representation of the group using the SchreierSims algorithm.

# Examples

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## Basic Examples(1)

A permutation group defined by two generators:

Compute its order:

## Scope(3)

An empty list of generators represents the identity (or trivial, or neutral) group:

Find the order of a group generated by two permutations:

Test the equality of permutation groups with the same support but possibly generated by different permutations:

They are different as Wolfram Language expressions:

## Applications(2)

This is the group of all rotations and reflections of a regular -sided polygon, the dihedral group, for . It can be generated by a rotation of an angle and a reflection along an axis through a vertex:

Construct the octagon corresponding to each group element:

This is the original polygon and its seven rotations. Numbers increase counterclockwise:

This is the polygon reflected along the bisection 15 and its seven rotations. Numbers increase clockwise:

The group of automorphisms of a graph is represented using PermutationGroup:

This is the number of automorphisms of the graph:

## Properties & Relations(3)

Explicit representation of a named group:

Generate the symmetric group of degree using transpositions:

Generate the alternating group of degree using generators:

## Neat Examples(1)

The moves of a Rubik's cube form a group. Number the moving facelets from 1 to 48:

These are the six basic rotations:

Group order:

Swapping two neighbor edge facelets is not allowed:

Simultaneous swaps of two edge pairs is allowed:

This is the superflip move, which switches all edge pairs simultaneously without changing any corner:

Edges and corners cannot be mixed (as the action of the group on the cube is not transitive), but any two corners or any two edges can be swapped:

Wolfram Research (2010), PermutationGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationGroup.html.

#### Text

Wolfram Research (2010), PermutationGroup, Wolfram Language function, https://reference.wolfram.com/language/ref/PermutationGroup.html.

#### CMS

Wolfram Language. 2010. "PermutationGroup." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PermutationGroup.html.

#### APA

Wolfram Language. (2010). PermutationGroup. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PermutationGroup.html

#### BibTeX

@misc{reference.wolfram_2024_permutationgroup, author="Wolfram Research", title="{PermutationGroup}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PermutationGroup.html}", note=[Accessed: 16-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_permutationgroup, organization={Wolfram Research}, title={PermutationGroup}, year={2010}, url={https://reference.wolfram.com/language/ref/PermutationGroup.html}, note=[Accessed: 16-September-2024 ]}