ConjugacyClasses

The conjugacy classes of a finite group — the orbits of $g \mapsto xgx^{-1}$. Their number is the dimension of the centre of $k[G]$.

WikipediaConjugacy class↗MathWorldConjugacyClass↗WikidataQ1353233↗nLabconjugacy class↗

ConjugacyClasses(group)the classes, as a list of lists

GroupCentreDimension(group)how many there are

Typed signatures

Type	Package	Applies
(expression<CyclicGroup> | expression<DihedralGroup> | expression<GroupDirectProduct>) -> list<list<expression<GroupBasis>>>	groupalgebra	always

Domain: Group algebras

Details

• An abelian group has one class per element, so $\dim Z(k[G]) = |G|$
• $D_n$ has $(n+3)/2$ classes for odd $n$ and $(n+6)/2$ for even $n$
• $D_3 \cong S_3$ has three: the identity, the transpositions, and the 3-cycles
• The classes partition the group, and the identity is always alone in its own

Examples

Basic #

abelian: one class per element

live ↗

In$\mathrm{GroupCentreDimension}(\mathrm{CyclicGroup}(6))$Out$6$

$D_3 \cong S_3$

live ↗

In$\mathrm{GroupCentreDimension}(\mathrm{DihedralGroup}(3))$Out$3$

Scope #

direct products work too

live ↗

In$\mathrm{GroupOrder}(\mathrm{GroupDirectProduct}(\mathrm{CyclicGroup}(2), \mathrm{CyclicGroup}(3)))$Out$6$

Properties #

$(4+6)/2$

live ↗

In$\mathrm{GroupCentreDimension}(\mathrm{DihedralGroup}(4))$Out$5$

About

Examples	4
Checked against	Wolfram Language 4 agreeSageMath 4 agreeOscar 4 agree
Crosswalk	WikipediaMathWorldWikidatanLab

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram has no conjugacy classes of a permutation group; GroupOrbits under the conjugation action would build them, but ours are lists of our own element labels (GroupBasis), which Wolfram does not have.

See also: ClassSum, GroupBasis