implemented (Inner)ActorCat1Group

CHANGES.md

@@ -1,6 +1,7 @@
  # CHANGES to the 'XMod' package
 
-## 2.92 -> 2.92dev (20/11/2024)
+## 2.92 -> 2.92dev (17/12/2024)
+ * (17/12/24) implemented (Inner)ActorCat1Group, fixing issue #144
  * (20/11/24) added AutomorphismPermGroup method for cat1-groups
               renamed PermAutomorphismAsXModMorphism as
               PermAutomorphismAs2dGroupMorphism and added cat1-group method

PackageInfo.g

@@ -8,7 +8,7 @@ SetPackageInfo( rec(
 PackageName := "XMod",
 Subtitle := "Crossed Modules and Cat1-Groups",
 Version := "2.92dev",
-Date := "28/11/2024", # dd/mm/yyyy format
+Date := "17/12/2024", # dd/mm/yyyy format
 License := "GPL-2.0-or-later",
 
 Persons := [
@@ -91,7 +91,7 @@ PackageDoc := rec(
   HTMLStart := "doc/chap0_mj.html",
   PDFFile   := "doc/manual.pdf",
   SixFile   := "doc/manual.six",
-  LongTitle := "Crossed Modules and Cat1-Groups in GAP",
+  LongTitle := "Crossed Modules and Cat1-Groups in GAP"
 ),
 
 Dependencies := rec(

doc/gp2act.xml

@@ -204,13 +204,14 @@ Crossed module Actor[c3->s3] :-
 
 The main methods for these operations are written for 
 permutation crossed modules. 
-For other crossed modules an isomorphism to a permutation crodssed module 
+For other crossed modules an isomorphism to a permutation crossed module 
 is found first, and then the main method is applied to the image. 
 In the example the crossed module <C>XAq8</C> 
 is the automorphism crossed module of the quaternion group. 
 <Example> 
 <![CDATA[
 gap> q8 := Group( (1,2,3,4)(5,8,7,6), (1,5,3,7)(2,6,4,8) );;
+gap> SetName( q8, "q8" );
 gap> XAq8 := XModByAutomorphismGroup( q8 );; 
 gap> StructureDescription( WhiteheadXMod( XAq8 ) ); 
 [ "Q8", "C2 x C2 x C2" ]
@@ -290,4 +291,85 @@ Crossed module InnerActor[c3->s3] :-
 
 </Section>
 
+<Section><Heading>Actor of a cat<M>^1</M>-group</Heading>
+
+<ManSection>
+   <Attr Name="ActorCat1Group"
+         Arg="cat1" />
+   <Func Name="InnerActorCat1Group"
+         Arg="cat1" />
+<Description>
+The actor of a cat<M>^1</M>-group <M>C</M> is obtained by converting
+<M>C</M> to a crossed module; forming the actor of that crossed module;
+and then converting that actor into a cat<M>^1</M>-group.
+<P/>
+A similar procedure is followed for the inner actor.
+</Description>
+</ManSection>
+<P/>
+<Example>
+<![CDATA[
+gap> C3;
+[g18 => s3]
+gap> AC3 := ActorCat1Group( C3 );
+cat1(Actor[c3->s3])
+gap> Display( AC3 );             
+Cat1-group cat1(Actor[c3->s3]) :- 
+: Source group has generators:
+  [ ( 9,10), ( 8, 9,10), ( 5, 7, 6)( 8, 9,10), (1,2)(3,4)(6,7)(8,9) ]
+: Range group has generators:
+  [ (5,7,6), (1,2)(3,4)(6,7) ]
+: tail homomorphism maps source generators to:
+  [ (), (), (5,7,6), (1,2)(3,4)(6,7) ]
+: head homomorphism maps source generators to:
+  [ (1,2)(3,4)(5,6), (5,7,6), (5,7,6), (1,2)(3,4)(6,7) ]
+: range embedding maps range generators to:
+  [ ( 5, 7, 6)( 8, 9,10), (1,2)(3,4)(6,7)(8,9) ]
+: kernel has generators:
+  [ ( 9,10), ( 8, 9,10) ]
+: boundary homomorphism maps generators of kernel to:
+  [ (1,2)(3,4)(5,6), (5,7,6) ]
+: kernel embedding maps generators of kernel to:
+  [ ( 9,10), ( 8, 9,10) ]
+: associated crossed module is Actor[c3->s3]
+gap> StructureDescription( AC3 );
+[ "S3 x S3", "S3" ]
+gap> IAC3 := InnerActorCat1Group( C3 );
+cat1(InnerActor[c3->s3])
+gap> StructureDescription( IAC3 );
+[ "(C3 x C3) : C2", "S3" ]
+]]>
+</Example>
+
+<ManSection>
+   <Func Name="Actor"
+         Arg="args" />
+   <Func Name="InnerActor"
+         Arg="args" />
+<Description>
+The global functions <C>Actor</C> and <C>InnerActor</C> 
+will call operations <C>ActorXMod</C> and <C>InnerActorXMod</C>
+or <C>ActorCat1Group</C> and <C>InnerActorCat1Group</C>
+as appropriate.
+</Description>
+</ManSection>
+<P/>
+<Example>
+<![CDATA[
+gap> c4q := Subgroup( q8, [ (1,2,3,4)(5,8,7,6) ] );;
+gap> SetName( c4q, "c4q" );                         
+gap> Xc4q := XModByNormalSubgroup( q8, c4q );;      
+gap> AXc4q := Actor( Xc4q );
+Actor[c4q->q8]
+gap> StructureDescription( AXc4q );
+[ "D8", "D8" ]
+gap> IAXc4q := InnerActor( Xc4q );
+InnerActor[c4q->q8]
+gap> StructureDescription( IAXc4q );
+[ "C2", "C2 x C2" ]
+]]>
+</Example>
+
+</Section>
+
 </Chapter>

lib/apps.gi

@@ -1,11 +1,11 @@
-##############################################################################
+############################################################################
 ##
-#W  apps.gi                    GAP4 package `XMod'               Chris Wensley
+#W  apps.gi                    GAP4 package `XMod'             Chris Wensley
 ##
 #Y  Copyright (C) 2001-2024, Chris Wensley et al,  
 #Y  School of Computer Science, Bangor University, U.K. 
 
-##############################################################################
+############################################################################
 ##
 InstallMethod( LoopClass, "for a crossed module and an element", true, 
     [ IsPreXMod, IsObject ], 0, 
@@ -39,7 +39,7 @@ function( X0, a )
 end ); 
 
 
-##############################################################################
+############################################################################
 ##
 InstallMethod( LoopClasses, "for a crossed module", true, [ IsPreXMod ], 0, 
 function( X0 ) 
@@ -179,7 +179,7 @@ function( X0, a )
     return XModByBoundaryAndAction( bdy1, act1 );  
 end ); 
 
-############################################################################## 
+############################################################################
 ## 
 InstallMethod( LoopClassesNew, "for a crossed module", true, 
     [ IsPreXMod ], 0, 
@@ -246,7 +246,7 @@ function( X0 )
     return classes; 
 end ); 
 
-############################################################################## 
+############################################################################
 ## 
 InstallMethod( LoopClassRepresentatives, "for a crossed module", true, 
     [ IsPreXMod ], 0, 
@@ -294,7 +294,7 @@ function( X0 )
     return reps; 
 end ); 
 
-############################################################################## 
+############################################################################
 ## 
 InstallMethod( LoopsXMod, "for a crossed module and an element", true, 
     [ IsPreXMod, IsObject ], 0, 
@@ -401,13 +401,13 @@ function( X0, a )
     return XModByBoundaryAndAction( bdy1, act1 );  
 end ); 
 
-##############################################################################
+############################################################################
 ##
 InstallMethod( AllLoopsXMod, "default method for a crossed module", true, 
     [ IsPreXMod ], 0, 
 function( X0 ) 
 
-    local bdy0, act0, M0, genM0, P0, genP0, igenP0, imgenM0, imbdy0, Q0, 
+    local bdy0, act0, M0, genM0, P0, genP0, igenP0, imbdy0, Q0, 
           conjP0, repsP0, eltsP0, nclP0, i, eclP0, relP0, c, p, genimbdy0, 
           len, class, j, r, k, q, pos, d, h, eqreps, numreps, a, X1, allX; 
 
@@ -418,7 +418,6 @@ function( X0 )
     P0 := Range( X0 ); 
     genP0 := GeneratorsOfGroup( P0 ); 
     igenP0 := List( genP0, g -> ImageElm( act0, g ) ); 
-    imgenM0 := List( genM0, g -> List( igenP0, p -> ImageElm(p,g) ) ); 
     imbdy0 := ImagesSource( bdy0 ); 
     Q0 := P0/imbdy0; 
     conjP0 := ConjugacyClasses( P0 ); 
@@ -427,7 +426,6 @@ function( X0 )
     nclP0 := Length( conjP0 ); 
     if ( InfoLevel( InfoXMod ) > 1 ) then 
         Print( "#I  Size(P0) = ", Size(P0), "\n" ); 
-        Print( "#I  imgenM0 = ", imgenM0, "\n" );   #? this is not used 
         Print( "#I  cokernel of boundary has IdGroup ", IdGroup(Q0), "\n" ); 
         Print( "#I  conjugacy class sizes, reps and orders:classes for P :-\n" ); 
         Print( "#I  ", List( conjP0, Size ), "\n" ); 

lib/gp2act.gd

@@ -53,10 +53,12 @@ DeclareAttribute( "ActorCat1Group", IsCat1Group );
 #############################################################################
 ##
 #A  InnerActorXMod( <XM> )
+#A  InnerActorCat1Group( <C> )
 #A  InnerMorphism( <XM> )
-#A  XModCentre( <XM> )     #? (18/07/11) : cannot get plain Centre to work ?? 
+#A  XModCentre( <XM> )     ## its not possible to use Centre for this 
 ##
 DeclareAttribute( "InnerActorXMod", IsXMod );
+DeclareAttribute( "InnerActorCat1Group", IsCat1Group );
 DeclareAttribute( "InnerMorphism", IsXMod );
 DeclareAttribute( "XModCentre", IsXMod ); 
 

lib/gp2act.gi

@@ -690,8 +690,9 @@ end );
 #############################################################################
 ##
 #M  ActorCat1Group( <C> )
+#M  InnerActorCat1Group( <C> )
 ##
-## a direct implementation might be better!
+#? direct implementations might be better!
 ##
 InstallMethod( ActorCat1Group, "actor cat1-group", true, [ IsCat1Group ], 0, 
 function( C )
@@ -703,6 +704,17 @@ function( C )
     return Cat1GroupOfXMod( AXC ); 
 end );
 
+InstallMethod( InnerActorCat1Group, "inner actor cat1-group", true,
+    [ IsCat1Group ], 0, 
+function( C )
+
+    local XC, IAXC;
+
+    XC := XModOfCat1Group( C );
+    IAXC := InnerActorXMod ( XC );
+    return Cat1GroupOfXMod( IAXC ); 
+end );
+
 #############################################################################
 ##
 #M  InnerMorphism( <XM> )
@@ -732,8 +744,6 @@ end );
 ##
 #M  XModCentre( <XM> )
 ##
-#?  InstallOtherMethod( Centre, "centre of an xmod", true, [ IsPermXMod ], 0, 
-##  
 InstallMethod( XModCentre, "centre of an xmod", true, [ IsXMod ], 0, 
 function( XM )
     return Kernel( InnerMorphism( XM ) );

lib/gp2ind.gi

@@ -446,7 +446,8 @@ function( X0, iota, trans )
     nrFM := Length( relFM );
 
     # determine  genN = closure of  genM  under conjugation by  P 
-    Info( InfoXMod, 2, "finding closure of GeneratorsOfGroup(M) by P-action");
+    Info( InfoXMod, 2, 
+        "finding closure of GeneratorsOfGroup(M) by P-action");
     genN := ShallowCopy( genM );
     ngN := Length( genN );
     genFN := ShallowCopy( genFM );
@@ -459,7 +460,7 @@ function( X0, iota, trans )
             posn := Position( genN, n2 );
             if ( posn = fail ) then
                 Add( genN, n2 );
-                m2 := ImageElm( g2fpM, n2 );## is this better?
+                m2 := ImageElm( g2fpM, n2 ); ## is this better?
                 if not ( n2 in M ) then 
                     Error( "M is not a normal subgroup of P" );
                 fi;
@@ -582,7 +583,7 @@ function( X0, iota, trans )
     Info( InfoXMod, 2, " genQ = ", genQ );
     Info( InfoXMod, 2, "geniN = ", geniN );
     Info( InfoXMod, 2, "  elQ = ", elQ );
-    Info( InfoXMod, 2, "imFIQ = images in Q of the generators of I: ", imFIQ );
+    Info( InfoXMod, 2, "imFIQ = images in Q of the gens of I: ", imFIQ );
     # Action of the generators of Q on the generators of I
     for j1 in [1..ngQ] do
         q1 := genQ[j1];
@@ -620,7 +621,7 @@ function( X0, iota, trans )
         t1 := T[i1];
         n1 := (i1-1)*ngN;
         Info( InfoXMod, 4, "[i1,t1] = ", [i1,t1] );
-        for j1 in [1..ngM] do                 #? no longer [1..ngN]
+        for j1 in [1..ngM] do
             m1 := genM[j1];
             t := ImageElm( comp, m1 )^t1;
             if not ( t in gimD ) then 
@@ -709,9 +710,8 @@ function( X0, iota, trans )
     free2 := FreeGroupOfFpGroup( FI2 );
     genfree2 := GeneratorsOfGroup( free2 );
     gensFI2 := GeneratorsOfPresentation( presFI1 );   ## ????? 
-    #? (10/07/10)  do we need both of genFI2 and gensFI2  ?????  
     Info( InfoXMod, 3, "gensFI2 = ", gensFI2 );
-    Info( InfoXMod, 3, "genFI2 = gensFI2 ? ", gensFI2 = gensFI2 );
+    Info( InfoXMod, 3, "genFI2 = gensFI2 ? ", genFI2 = gensFI2 );
     Info( InfoXMod, 1, "#I induced group has size: ", oFI2 );
     #? (19/07/11) : example InducedXMod( s4, s3b, s3b ) fails 
     #? because of a pc isomorphism instead of a perm isomorphism,
@@ -842,7 +842,7 @@ function( X0, iota, trans )
         Print( "mor: X0 -> IX  not an xmod morphism!\n" );
     fi;
     if IsPermGroup( IX ) then 
-        sdpr := SmallerDegreePermutationRepresentation2DimensionalGroup( IX );
+        sdpr := SmallerDegreePermutationRepresentation2DimensionalGroup(IX);
         if not ( sdpr = fail ) then 
             IX := Range( sdpr );
             mor := mor * sdpr;

tst/extra/induced.tst

@@ -1,6 +1,6 @@
-##############################################################################
+############################################################################
 ##
-#W  induced.tst                 GAP4 package `XMod'              Chris Wensley
+#W  induced.tst                 GAP4 package `XMod'            Chris Wensley
 ##  
 #Y  Copyright (C) 2001-2023, Chris Wensley et al, 
 #Y  School of Computer Science, Bangor University, U.K. 
@@ -293,6 +293,6 @@ gap> StructureDescription( indXs3c );
 gap> SetInfoLevel( InfoXMod, saved_infolevel_xmod );; 
 gap> STOP_TEST( "induced.tst", 10000 );
 
-##############################################################################
+#############################################################################
 ##
-#E  induced.tst . . . . . . . . . . . . . . . . . . . . . . . . . . ends here
+#E  induced.tst . . . . . . . . . . . . . . . . . . . . . . . . .  ends here

tst/manual/apps.tst

@@ -1,6 +1,6 @@
-##############################################################################
+#############################################################################
 ##
-#W  apps.tst                    GAP4 package `XMod'              Chris Wensley
+#W  apps.tst                    GAP4 package `XMod'             Chris Wensley
 ##  
 #Y  Copyright (C) 2001-2022, Chris Wensley et al, 
 #Y  School of Computer Science, Bangor University, U.K.