gap> ColorPrompt(true); gap> 1/0; Error, Rational operations: must not be zero not in any function at *stdin*:3 you can replace via 'return ;' brk> ColorPrompt(false); brk> 1/0; Error, Rational operations: must not be zero not in any function at *errin*:2 you can replace via 'return ;' brk_2> quit; brk> quit; gap> LoadPackage("laguna"); true gap> G := DihedralGroup(16); gap> ZG := GroupRing(Integers, G); gap> QG := GroupRing(Rationals, G); gap> FG := GroupRing(GF(2), G); gap> IsGroupAlgebra(ZG); false gap> IsGroupAlgebra(QG); true gap> i := Embedding(G, ZG); FLMLORWithOne( Integers, ... ) > gap> z := MinimalGeneratingSet(Center(G))[1]; f4 gap> z in ZG; false gap> MinimalGeneratingSet(Center(G)); [ f4 ] gap> z := MinimalGeneratingSet(Center(G))[1]; f4 gap> z^i in ZG; true gap> a := First(Elements(G), g -> Order(g) = 8); f2 gap> v := z^i + a^i; (1)*f2+(1)*f4 gap> IsSymmetric(v); false gap> IsSymmetric(v+Involution(v)); true gap> IsSymmetric(v*Involution(v)); true gap> Augmentation(v); 2 gap> v^-1; fail gap> b := First(Elements(G), g -> Order(g) = 2 and not g in Center(G)); f1 gap> x := v - b^i; (-1)*f1+(1)*f2+(1)*f4 gap> Augmentation(x); 1 gap> x^-1; (1/3)* of ...+(-1/3)*f1+(1/3)*f2+(1/3)*f3+(1/3)*f4+(-1/3)*f1*f2+(-1/ 3)*f1*f3+(2/3)*f1*f4+(1/3)*f2*f3+(-2/3)*f2*f4+(-2/3)*f3*f4+(2/3)*f1*f2*f3+(2/ 3)*f1*f2*f4+(-1/3)*f1*f3*f4+(1/3)*f2*f3*f4+(-1/3)*f1*f2*f3*f4 gap> x^-1 in ZG; false gap> x^-1 in QG; false gap> j := Embedding(G, QG); AlgebraWithOne( Rationals, ... ) > gap> x := a^j + z^j - b^j; (-1)*f1+(1)*f2+(1)*f4 gap> x^-1 in QG; true gap> BicyclicUnitOfType(a^i, b^i); Error, Variable: 'BicyclicUnitOfType' must have a value not in any function at *stdin*:32 gap> BicyclicUnitOfType1(a^i, b^i); (1)* of ... gap> u1 := BicyclicUnitOfType2(a^i, b^i); (1)* of ... gap> u1 := BicyclicUnitOfType2(b^i, a^i); (1)* of ...+(-1)*f2+(-1)*f1*f2+(1)*f2*f3*f4+(1)*f1*f2*f3*f4 gap> u := u1*b^i; (1)*f1+(1)*f2+(1)*f1*f2+(-1)*f2*f3*f4+(-1)*f1*f2*f3*f4 gap> Order(u); 2 gap> BassCyclicUnit(a^i, 3); (-8)* of ...+(-6)*f2+(9)*f4+(6)*f2*f3+(6)*f2*f4+(-6)*f2*f3*f4 gap> LoadPackage("wedderga"); #I You may wish to install the xgap package #I and enjoy the graphic capabilities of SONATA. ___________________________________________________________________________ / ___ || / \ /\ Version2.8 || || || |\ | / \ /\ Erhard Aichinger \___ || || |\\ | /____\_____________/__\ Franz Binder \ || || | \\ | / \ || / \ Juergen Ecker || \___/ | \\ | / \ || / \ Peter Mayr || | \\| / \ || Christof Noebauer \___/ | \| || System Of Nearrings And Their Applications Info: http://www.algebra.uni-linz.ac.at/Sonata/ ____ | / \ / --+-- Version 3.14 / | | |\\ //| | | _ | | | \\ // | the GUAVA Group | \ | | |--\\ //--| \ || | | \\ // | \___/ \___/ | \\// | ───────────────────────────────────────────────────────────────────────────── Loading Wedderga 4.9.3 (Wedderga) by Gurmeet Kaur Bakshi (gkbakshi@pu.ac.in), Osnel Broche Cristo (osnel@ufla.br), Allen Herman (aherman@math.uregina.ca), Alexander Konovalov (https://alexk.host.cs.st-andrews.ac.uk), Sugandha Maheshwary (sugandha@iisermohali.ac.in), Gabriela Olteanu (http://math.ubbcluj.ro/~olteanu), Aurora Olivieri (olivieri@usb.ve), Angel del Rio (http://www.um.es/adelrio), and Inneke Van Gelder (http://homepages.vub.ac.be/~ivgelder). Homepage: https://gap-packages.github.io/wedderga ───────────────────────────────────────────────────────────────────────────── true gap> WedderburnDecomposition(GroupRing(Rationals, CyclicGroup(3))); [ Rationals, CF(3) ] gap> WedderburnDecompositionInfo(GroupRing(Rationals, CyclicGroup(3))); [ [ 1, Rationals ], [ 1, CF(3) ] ] gap> WedderburnDecompositionInfo(GroupRing(GF(5), CyclicGroup(3))); [ [ 1, 5 ], [ 1, 25 ] ] gap> WedderburnDecompositionInfo(GroupRing(GF(7), CyclicGroup(3))); [ [ 1, 7 ], [ 1, 7 ], [ 1, 7 ] ] gap> WedderburnDecomposition(GroupRing(Rationals, QuaternionGroup(8))); [ Rationals, Rationals, Rationals, Rationals, ] gap> WedderburnDecompositionInfo(GroupRing(Rationals, QuaternionGroup(8))); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 4, [ 2, 3, 2 ] ] ] gap> ?WedderburnDecompositionInfo Help: Showing `Wedderga: #[22m#[34mWedderburnDecompositionInfo#[0m' gap> SetHelpViewer("firefox"); gap> ?WedderburnDecompositionInfo Help: Showing `Wedderga: #[22m#[34mWedderburnDecompositionInfo#[0m' gap> SetHelpViewer("screen"); gap> WedderburnDecompositionWithDivAlgParts(GroupRing(Rationals, QuaternionGroup(8))); [ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, rec( Center := Rationals, DivAlg := true, LocalIndices := [ [ 2, 2 ], [ infinity, 2 ] ], SchurIndex := 2 ) ] ] gap> S3 := SymmetricGroup(3); Sym( [ 1 .. 3 ] ) gap> QS3 := GroupRing(Rationals, S3); gap> idems := PrimitiveCentralIdempotentsByCharacterTable(QS3); [ (1/6)*()+(-1/6)*(2,3)+(-1/6)*(1,2)+(1/6)*(1,2,3)+(1/6)*(1,3,2)+(-1/6)*(1,3), (2/3)*()+(-1/3)*(1,2,3)+(-1/3)*(1,3,2), (1/6)*()+(1/6)*(2,3)+(1/6)*(1,2)+(1/ 6)*(1,2,3)+(1/6)*(1,3,2)+(1/6)*(1,3) ] gap> IsCompleteSetOfOrthogonalIdempotents(idems); Error, no method found! For debugging hints type ?Recovery from NoMethodFound Error, no 1st choice method found for `IsCompleteSetOfOrthogonalIdempotents' o\ n 1 arguments at /home/andreas/.opt/gap-4.9.3/lib/methsel2.g:250 called from ( ) called from read-eval loop at *stdin*:54 you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk> quit; gap> IsCompleteSetOfOrthogonalIdempotents(QS3, idems); true gap> IsCompleteSetOfOrthogonalIdempotents(QS3, [One(QS3)]); true gap> LogTo();