GroupOrder

$|G|$: how many elements a group has. Works over every carrier this package knows — CyclicGroup, DihedralGroup, GroupDirectProduct, SymmetricGroup, AlternatingGroup and PermutationGroup.

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

GroupOrder(group)the number of elements of group

Oscarorder((group).G)

Typed signatures

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

Domain: Permutations

Details

• For a PermutationGroup, the order comes from a breadth-first closure over its generators, not a stored table
• GroupOrder(SymmetricGroup(n)) is $n!$, and GroupOrder(AlternatingGroup(n)) is $n!/2$, without materialising the permutations

Examples

Basic #

live ↗

In$\mathrm{GroupOrder}(\mathrm{DihedralGroup}(100))$Out$200$

Order of a group generated by two permutations of 11 points

live ↗

In$\mathrm{GroupOrder}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigl\lbrack2, 10\bigr\rbrack, \bigl\lbrack4, 11\bigr\rbrack, \bigl\lbrack5, 7\bigr\rbrack, \bigl\lbrack8, 9\bigr\rbrack\bigr\rbrack), \mathrm{Cycles}(\bigl\lbrack\bigl\lbrack1, 4, 3, 8\bigr\rbrack, \bigl\lbrack2, 5, 6, 7\bigr\rbrack\bigr\rbrack)\bigr\rbrack))$Out$2\,880$

Scope #

live ↗

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

$4!$, without unranking every permutation

live ↗

In$\mathrm{GroupOrder}(\mathrm{SymmetricGroup}(4))$Out$24$

Properties #

No generators gives the trivial group of order 1

live ↗

In$\mathrm{GroupOrder}(\mathrm{PermutationGroup}(\bigl\lbrack\bigr\rbrack))$Out$1$

The identity alone also generates a group of order 1

live ↗

In$\mathrm{GroupOrder}(\mathrm{PermutationGroup}(\bigl\lbrack\mathrm{Cycles}(\bigl\lbrack\bigr\rbrack)\bigr\rbrack))$Out$1$

About

Examples	6
Checked against	Wolfram Language 6 agreeOscar 1 agree
Crosswalk	Wolfram LanguageOscar

Implementation

mappedoscarpackages/oracle/src/mappings.ts

See also: GroupElements, PermutationGroup, DihedralGroup, CyclicGroup, SymmetricGroup, AlternatingGroup