GroupGenerators

The generating permutations a PermutationGroup was built from, in Cycles notation — or, for AlternatingGroup, the fixed pair Wolfram itself uses.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageGroupGenerators✓↗

GroupGenerators(group)the generators passed to PermutationGroup, unchanged; or AlternatingGroup(n)'s canonical pair

SageMathenumeratio_group_generators(group)

Typed signatures

Type	Package	Applies
(expression<AlternatingGroup> | expression<PermutationGroup>) -> list<expression<Cycles>>	groupalgebra	always

Domain: Permutations

Examples

Basic #

live ↗

In$\mathrm{GroupGenerators}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 9, 6\bigr\rbrack, \bigl\lbrack3, 7\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 9, 6\bigr\rbrack, \bigl\lbrack3, 7\bigr\rbrack\bigr\rbrack)\bigr\rbrack$

A group built from generators returns them as given

live ↗

In$\mathrm{GroupGenerators}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack\bigr\rbrack)\bigr\rbrack$

Scope #

Generators of the Klein four-group, kept in order

live ↗

In$\mathrm{GroupGenerators}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2\bigr\rbrack, \bigl\lbrack3, 4\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 3\bigr\rbrack, \bigl\lbrack2, 4\bigr\rbrack\bigr\rbrack)\bigr\rbrack$

About

Examples	3
Checked against	Wolfram Language 3 agreeSageMath 3 agree
Crosswalk	Wolfram LanguageSageMath

Implementation

mappedsagepackages/oracle/src/mappings.ts

Sage chooses its own generators for a named group (AlternatingGroup(5) is generated by a 3-cycle and the 5-cycle in the other order), so only a PermutationGroup's own generators are a like-for-like comparison.

See also: PermutationGroup, AlternatingGroup, GroupOrder, GroupElements