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FIX: Add IsBiCoset #2666

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Aug 7, 2018
Merged

FIX: Add IsBiCoset #2666

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45f14a7
FIX: Add IsBiCoset
hulpke Jul 28, 2018
9d4cc52
REBASE: typo
hulpke Jul 31, 2018
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1 change: 1 addition & 0 deletions doc/ref/groups.xml
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Original file line number Diff line number Diff line change
Expand Up @@ -175,6 +175,7 @@ free generators, that will map on this element again.
<#Include Label="RightCosets">
<#Include Label="CanonicalRightCosetElement">
<#Include Label="IsRightCoset">
<#Include Label="IsBiCoset">
<#Include Label="CosetDecomposition">

</Section>
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20 changes: 20 additions & 0 deletions lib/csetgrp.gd
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Expand Up @@ -320,6 +320,24 @@ DeclareAttribute( "RepresentativesContainedRightCosets", IsDoubleCoset );
DeclareCategory("IsRightCoset", IsDomain and IsExternalOrbit and
IsMultiplicativeElementWithInverse);

#############################################################################
##
#P IsBiCoset( <C> )
##
## <#GAPDoc Label="IsBiCoset">
## <ManSection>
## <Prop Name="IsBiCoset" Arg='C'/>
##
## <Description>
## A (right) cose is considered a <E>bicoset</E> if its set of elements simultaneously forms a left
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Typo: coset

## coset for the same subgroup. This is the case, for example, if the coset representative normalizes
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Why "for example"? This is an equivalence, isn't it?

## the subgroup.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareProperty( "IsBiCoset", IsRightCoset );

#############################################################################
##
#O RightCoset( <U>, <g> )
Expand Down Expand Up @@ -353,6 +371,8 @@ DeclareCategory("IsRightCoset", IsDomain and IsExternalOrbit and
## 6
## gap> AsList(c);
## [ (2,3,4), (1,4,2), (1,3,4,2), (1,3)(2,4), (2,4), (1,4,2,3) ]
## gap> IsBiCoset(c);
## false
## ]]></Example>
## </Description>
## </ManSection>
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27 changes: 22 additions & 5 deletions lib/csetgrp.gi
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Expand Up @@ -629,6 +629,14 @@ function(d)
Print(ViewString(d));
end);

InstallMethod(IsBiCoset,"test property",true,[IsRightCoset],0,
function(c)
local s,r;
s:=ActingDomain(c);
r:=Representative(c);
return ForAll(GeneratorsOfGroup(s),x->r*x/r in s);
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Perhaps test for x^r in s instead, as that will be a bit more efficient in some cases (e.g. for permutations)?

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Yes, if this ever could be time-critical this has the potential to be faster. I decided on the inverse-conjugation as it translated directly to the condition as I had written it doen on paper.

end);

InstallMethodWithRandomSource(Random,
"for a random source and a RightCoset",
[IsRandomSource, IsRightCoset],0,
Expand Down Expand Up @@ -656,10 +664,17 @@ end);
InstallOtherMethod(\*,"RightCosets",IsIdenticalObj,
[IsRightCoset,IsRightCoset],0,
function(a,b)
if ActingDomain(a)<>ActingDomain(b) then
Error("no multiplication defined for cosets of different subgroups");
local c;
if ActingDomain(a)<>ActingDomain(b) then TryNextMethod();fi;
if not IsBiCoset(a) then # product does not require b to be bicoset
TryNextMethod();
fi;
c:=RightCoset(ActingDomain(a), Representative(a) * Representative(b) );
if IsBiCoset(b) then
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Perhaps first check HasIsBicoset to avoid a computation?

SetIsBiCoset(c,true);
fi;
return RightCoset(ActingDomain(a), Representative(a) * Representative(b) );

return c;
end);

InstallOtherMethod(InverseOp,"Right cosets",true,
Expand All @@ -668,10 +683,12 @@ function(a)
local s,r;
s:=ActingDomain(a);
r:=Representative(a);
if ForAny(GeneratorsOfGroup(s),x->not x^r in s) then
if not IsBiCoset(a) then
Error("Inversion only works for cosets of normal subgroups");
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That error message is wrong (though of course it was before, too)

fi;
return RightCoset(s,Inverse(r));
r:=RightCoset(s,Inverse(r));
SetIsBiCoset(r,true);
return r;
end);

InstallOtherMethod(OneOp,"Right cosets",true,
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