GroupBasis

A basis element of the group algebra $k[G]$, named by its group element's label. The product is the group's own multiplication, extended bilinearly.

WikipediaGroup ring↗MathWorldGroupRing↗WikidataQ2602722↗

GroupBasis(label)the basis element for that group element

SageMathenumeratio_basis(label)OscarEnumeratioBasis(label)

GroupProduct(group, a, b)multiply — the group is named, since a basis element does not carry it

Typed signatures

Type	Package	Applies
(string) -> number	groupalgebra	always

Domain: Group algebras

Details

• Groups: CyclicGroup(n) with elements $0 \dots n-1$, DihedralGroup(n) with elements k for $r^k$ and s k for $s r^k$, and GroupDirectProduct(g, h)
• $k[\mathbb{Z}_n]$ multiplies by adding indices mod $n$ — it is $k[x]/(x^n-1)$
• $k[G]$ is commutative exactly when $G$ is abelian
• Dihedral relations: $r^n = 1$, $s^2 = 1$, $s r s = r^{-1}$
• $\dim k[G] = |G|$, and Basis(GroupAlgebra(group)) lists the elements

Examples

Basic #

$2 + 5 = 7 \equiv 1$

live ↗

In$\mathrm{GroupProduct}(\mathrm{CyclicGroup}(6), \mathrm{GroupBasis}(\text{2}), \mathrm{GroupBasis}(\text{5}))$Out$\mathrm{GroupBasis}(\text{1})$

$|D_4| = 8$

live ↗

In$\mathrm{AlgebraDimension}(\mathrm{GroupAlgebra}(\mathrm{DihedralGroup}(4)))$Out$8$

Properties #

$s^2 = 1$

live ↗

In$\mathrm{GroupProduct}(\mathrm{DihedralGroup}(4), \mathrm{GroupBasis}(\text{s0}), \mathrm{GroupBasis}(\text{s0}))$Out$\mathrm{GroupBasis}(\text{0})$

so $k[D_4]$ is not commutative either

live ↗

In$\mathrm{GroupIsAbelian}(\mathrm{DihedralGroup}(4))$Out$\mathrm{False}$

About

Examples	4
Checked against	SageMath 4 agreeOscar 4 agreeWolfram Language 1 agree
Crosswalk	WikipediaMathWorldWikidataSageMathOscar

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedwolframpackages/oracle/src/mappings.ts

A basis element is a label of our group algebra's own elements; Wolfram has no group algebra.

See also: ClassSum, ConjugacyClasses, Basis