GroupElements

Every element of a group, or — for a PermutationGroup or AlternatingGroup with a second, list argument — the elements at those positions (Part-style, negative counts from the end).

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GroupElements(group)all of group's elements

GroupElements(group, positions)a PermutationGroup's or AlternatingGroup's elements at positions, in its own order

Typed signatures

Type	Package	Applies
(expression<AlternatingGroup> | expression<CyclicGroup> | expression<DihedralGroup> | expression<GroupDirectProduct> | expression<PermutationGroup>, list<integer>?) -> list<expression>	groupalgebra	always

Domain: Permutations

Details

• A PermutationGroup's or AlternatingGroup's elements print as Cycles, not as GroupBasis labels — its elements are permutations, not abstract group-algebra basis vectors
• The identity always comes first; the rest follow the ascending order of their one-line words

Examples

Basic #

live ↗

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

Scope #

live ↗

In$\mathrm{GroupElements}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 9, 6\bigr\rbrack, \bigl\lbrack3, 7\bigr\rbrack\bigr\rbrack)\bigr\rbrack), \bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack3, 7\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 6, 9\bigr\rbrack\bigr\rbrack)\bigr\rbrack$

a negative position counts from the end

live ↗

In$\mathrm{GroupElements}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 9, 6\bigr\rbrack, \bigl\lbrack3, 7\bigr\rbrack\bigr\rbrack)\bigr\rbrack), \bigl\lbrack-1\bigr\rbrack)$Out$\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 9, 6\bigr\rbrack, \bigl\lbrack3, 7\bigr\rbrack\bigr\rbrack)\bigr\rbrack$

live ↗

In$\mathrm{Same}(\mathrm{GroupElements}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 2, 3\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack2, 3, 4\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack3, 4, 5\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack4, 5, 6\bigr\rbrack\bigr\rbrack)\bigr\rbrack)), \mathrm{GroupElements}(\mathrm{AlternatingGroup}(6)))$Out$\mathrm{True}$

About

Examples	4
Checked against	Wolfram Language 4 agree
Crosswalk	Wolfram Language

See also: GroupOrder, PermutationGroup, AlternatingGroup, GroupGenerators