# Group Algebras

- [Introduction](introduction.md)

- [Construction of Group Algebras and their Elements](creation.md)

  - [Construction of a Group Algebra](creation.md#construction-of-a-group-algebra)

    - [`GroupAlgebra( R, G: parameters ): Rng, Grp → AlgGrp`](creation.md#function-groupalgebra-rng-grp)

    - [`GroupAlgebra< R, G: parameters >: Rng, Grp → AlgGrp`](creation.md#constructor-constructor-groupalgebra-rng-grp-alggrp)

    - [`Example: creation`](creation.md#example-ex-a0675c)

  - [Construction of a Group Algebra Element](creation.md#construction-of-a-group-algebra-element)

    - [`elt< A | r, g >: AlgGrp, RngElt, GrpElt → AlgGrpElt`](creation.md#constructor-constructor-elt-alggrp-rngelt-grpelt-alggrpelt)

    - [`A ! g: AlgGrp, GrpElt → AlgGrpElt`](creation.md#operation-op-alggrp-grpelt)

    - [`A ! r: AlgGrp, RngElt → AlgGrpElt`](creation.md#operation-op-alggrp-rngelt)

    - [`A ! [c₁, ..., cₙ]: AlgGrp, SeqEnum → AlgGrpElt`](creation.md#operation-op-alggrp-seqenum)

    - [`Eta(A): AlgGrp → AlgGrpElt`](creation.md#function-eta-alggrp)

    - [`Example: El Creation`](creation.md#example-ex-b66604)

- [Construction of Subalgebras, Ideals and Quotient Algebras](subring-ideal-quotient.md)

  - [`sub< cat : A | L>: Cat, AlgGrp, List → AlgGrp, Map`](subring-ideal-quotient.md#constructor-constructor-sub-cat-alggrp-list-alggrp-map)

  - [`lideal< cat : A | L>: Cat, AlgGrp, List → AlgGrp, Map`](subring-ideal-quotient.md#constructor-constructor-lideal-cat-alggrp-list-alggrp-map)

  - [`rideal< cat : A | L>: Cat, AlgGrp, List → AlgGrp, Map`](subring-ideal-quotient.md#constructor-constructor-rideal-cat-alggrp-list-alggrp-map)

  - [`ideal< cat : A | L>: Cat, AlgGrp, List → AlgGrp, Map`](subring-ideal-quotient.md#constructor-constructor-ideal-cat-alggrp-list-alggrp-map)

  - [`a * I: AlgGrpElt, AlgGrpSub → AlgGrpSub`](subring-ideal-quotient.md#operation-op-times-alggrpelt-alggrpsub)

  - [`I * a: AlgGrpSub, AlgGrpElt → AlgGrpSub`](subring-ideal-quotient.md#operation-op-times-alggrpsub-alggrpelt)

  - [`quo< A | L >: AlgGrp, List → AlgAss, Map`](subring-ideal-quotient.md#constructor-constructor-quo-alggrp-list-algass-map)

  - [`A / S: AlgGrp, AlgGrpSub → AlgAss`](subring-ideal-quotient.md#operation-op-div-alggrp-alggrpsub)

  - [`Example: wedderburn`](subring-ideal-quotient.md#example-ex-d83263)

- [Operations on Group Algebras and their Subalgebras](grp-alg-ops.md)

  - [Operations on Group Algebras](grp-alg-ops.md#operations-on-group-algebras)

    - [`Algebra(A): AlgGrp → AlgAss, Map`](grp-alg-ops.md#function-algebra-alggrp)

    - [`AugmentationMap(A): AlgGrp → Map`](grp-alg-ops.md#function-augmentationmap-alggrp)

    - [`AugmentationIdeal(A): AlgGrp → AlgGrpSub`](grp-alg-ops.md#function-augmentationideal-alggrp)

    - [`RepresentationType(A): AlgGrp → MonStgElt`](grp-alg-ops.md#function-representationtype-alggrp)

    - [`ChangeRepresentationType(A, Rep): AlgGrp, MonStgElt → AlgGrp, Map`](grp-alg-ops.md#function-changerepresentationtype-alggrp-monstgelt)

    - [`ConstructTable(A): AlgGrp`](grp-alg-ops.md#function-constructtable-alggrp)

    - [`CoefficientRing(A): AlgGrp → Rng`](grp-alg-ops.md#function-coefficientring-alggrp)

    - [`BaseRing(A): AlgGrp → Rng`](grp-alg-ops.md#function-basering-alggrp)

  - [Operations on Subalgebras of Group Algebras](grp-alg-ops.md#operations-on-subalgebras-of-group-algebras)

    - [`S ! 1: AlgGrpSub → Grp`](grp-alg-ops.md#operation-op-alggrpsub)

    - [`S ! 1: AlgGrp → Grp`](grp-alg-ops.md#operation-op-alggrp)

    - [`Group(S): AlgGrpSub → Grp`](grp-alg-ops.md#function-group-alggrpsub)

    - [`Group(S): AlgGrp → Grp`](grp-alg-ops.md#function-group-alggrp)

    - [`GroupAlgebra(S): AlgGrpSub → AlgGrp`](grp-alg-ops.md#function-groupalgebra-alggrpsub)

    - [`GroupAlgebra(S): AlgGrp → AlgGrp`](grp-alg-ops.md#function-groupalgebra-alggrp)

    - [`Module(S): AlgGrpSub → ModTupRng, Map`](grp-alg-ops.md#function-module-alggrpsub)

    - [`Module(S): AlgGrp → ModTupRng, Map`](grp-alg-ops.md#function-module-alggrp)

    - [`CoefficientRing(A): AlgGrpSub → Rng`](grp-alg-ops.md#function-coefficientring-alggrpsub)

    - [`BaseRing(A): AlgGrpSub → Rng`](grp-alg-ops.md#function-basering-alggrpsub)

    - [`BasisMatrix(S): AlgGrpSub → ModMatRngElt`](grp-alg-ops.md#function-basismatrix-alggrpsub)

    - [`BasisMatrix(S): AlgGrp → ModMatRngElt`](grp-alg-ops.md#function-basismatrix-alggrp)

    - [`Coordinates(S, a): AlgGrpSub, AlgGrpElt → [ RingElt ]`](grp-alg-ops.md#function-coordinates-alggrpsub-alggrpelt)

    - [`Coordinates(S, a): AlgGrp, AlgGrpElt → [ RingElt ]`](grp-alg-ops.md#function-coordinates-alggrp-alggrpelt)

    - [`IsLeftIdeal(S): AlgGrpSub → BoolElt`](grp-alg-ops.md#function-isleftideal-alggrpsub)

    - [`IsLeftIdeal(S): AlgGrp → BoolElt`](grp-alg-ops.md#function-isleftideal-alggrp)

    - [`IsRightIdeal(S): AlgGrpSub → BoolElt`](grp-alg-ops.md#function-isrightideal-alggrpsub)

    - [`IsRightIdeal(S): AlgGrp → BoolElt`](grp-alg-ops.md#function-isrightideal-alggrp)

    - [`IsIdeal(S): AlgGrpSub → BoolElt`](grp-alg-ops.md#function-isideal-alggrpsub)

    - [`IsIdeal(S): AlgGrp → BoolElt`](grp-alg-ops.md#function-isideal-alggrp)

    - [`Centraliser(S): AlgGrpSub → AlgGrpSub`](grp-alg-ops.md#function-centraliser-alggrpsub)

    - [`Centralizer(S): AlgGrpSub → AlgGrpSub`](grp-alg-ops.md#function-centralizer-alggrpsub)

    - [`Centraliser(S): AlgGrp → AlgGrpSub`](grp-alg-ops.md#function-centraliser-alggrp)

    - [`Centralizer(S): AlgGrp → AlgGrpSub`](grp-alg-ops.md#function-centralizer-alggrp)

    - [`Idealiser(S): AlgGrpSub → AlgGrpSub`](grp-alg-ops.md#function-idealiser-alggrpsub)

    - [`Idealizer(S): AlgGrpSub → AlgGrpSub`](grp-alg-ops.md#function-idealizer-alggrpsub)

    - [`Idealiser(S): AlgGrp → AlgGrpSub`](grp-alg-ops.md#function-idealiser-alggrp)

    - [`Idealizer(S): AlgGrp → AlgGrpSub`](grp-alg-ops.md#function-idealizer-alggrp)

    - [`LeftAnnihilator(S): AlgGrpSub → AlgGrpSub`](grp-alg-ops.md#function-leftannihilator-alggrpsub)

    - [`LeftAnnihilator(S): AlgGrp → AlgGrpSub`](grp-alg-ops.md#function-leftannihilator-alggrp)

    - [`RightAnnihilator(S): AlgGrpSub → AlgGrpSub`](grp-alg-ops.md#function-rightannihilator-alggrpsub)

    - [`RightAnnihilator(S): AlgGrp → AlgGrpSub`](grp-alg-ops.md#function-rightannihilator-alggrp)

    - [`Example: jacobson`](grp-alg-ops.md#example-ex-71b601)

- [Operations on Elements](elt-ops.md)

  - [`a + r: AlgGrpElt, RngElt → AlgGrpElt`](elt-ops.md#operation-op-plus-alggrpelt-rngelt)

  - [`r + a: RngElt, AlgGrpElt → AlgGrpElt`](elt-ops.md#operation-op-plus-rngelt-alggrpelt)

  - [`a + g: AlgGrpElt, GrpElt → AlgGrpElt`](elt-ops.md#operation-op-plus-alggrpelt-grpelt)

  - [`g + a: GrpElt, AlgGrpElt → AlgGrpElt`](elt-ops.md#operation-op-plus-grpelt-alggrpelt)

  - [`a - r: AlgGrpElt, RngElt → AlgGrpElt`](elt-ops.md#operation-op-minus-alggrpelt-rngelt)

  - [`r - a: RngElt, AlgGrpElt → AlgGrpElt`](elt-ops.md#operation-op-minus-rngelt-alggrpelt)

  - [`a - g: AlgGrpElt, GrpElt → AlgGrpElt`](elt-ops.md#operation-op-minus-alggrpelt-grpelt)

  - [`g - a: GrpElt, AlgGrpElt → AlgGrpElt`](elt-ops.md#operation-op-minus-grpelt-alggrpelt)

  - [`a * r: AlgGrpElt, RngElt → AlgGrpElt`](elt-ops.md#operation-op-times-alggrpelt-rngelt)

  - [`r * a: RngElt, AlgGrpElt → AlgGrpElt`](elt-ops.md#operation-op-times-rngelt-alggrpelt)

  - [`g * a: GrpElt, AlgGrpElt → AlgGrpElt`](elt-ops.md#operation-op-times-grpelt-alggrpelt)

  - [`a * g: AlgGrpElt, GrpElt → AlgGrpElt`](elt-ops.md#operation-op-times-alggrpelt-grpelt)

  - [`Support(a): AlgGrpElt → SeqEnum`](elt-ops.md#function-support-alggrpelt)

  - [`Trace(a): AlgGrpElt → RngElt`](elt-ops.md#function-trace-alggrpelt)

  - [`Augmentation(a): AlgGrpElt → RngElt`](elt-ops.md#function-augmentation-alggrpelt)

  - [`Involution(a): AlgGrpElt → AlgGrpElt`](elt-ops.md#function-involution-alggrpelt)

  - [`Coefficient(a, g): AlgGrpElt, GrpElt → RngElt`](elt-ops.md#function-coefficient-alggrpelt-grpelt)

  - [`a[g]: AlgGrpElt, GrpElt → RngElt`](elt-ops.md#indexed-indexed-a-alggrpelt-grpelt-rngelt)

  - [`ElementToSequence(a): AlgGrpElt → SeqEnum`](elt-ops.md#function-elementtosequence-alggrpelt)

  - [`Eltseq(a): AlgGrpElt → SeqEnum`](elt-ops.md#function-eltseq-alggrpelt)

  - [`Coefficients(a): AlgGrpElt → SeqEnum`](elt-ops.md#function-coefficients-alggrpelt)

  - [`Centraliser(a): AlgGrpElt → AlgGrpSub`](elt-ops.md#function-centraliser-alggrpelt)

  - [`Centralizer(a): AlgGrpElt → AlgGrpSub`](elt-ops.md#function-centralizer-alggrpelt)

  - [`Centraliser(S, a): AlgGrpSub, AlgGrpElt → AlgGrpSub`](elt-ops.md#function-centraliser-alggrpsub-alggrpelt)

  - [`Centralizer(S, a): AlgGrpSub, AlgGrpElt → AlgGrpSub`](elt-ops.md#function-centralizer-alggrpsub-alggrpelt)

  - [`Example: powering`](elt-ops.md#example-ex-76435e)

  - [`Example: average`](elt-ops.md#example-ex-1a2ff9)