Merge pull request #149 from gap-packages/issues

changed examples of induced crossed modules
gap-packages · Jan 3, 2025 · d2d3fa1 · d2d3fa1
2 parents ee8d4ba + 7e1c510
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diff --git a/CHANGES.md b/CHANGES.md
@@ -1,6 +1,7 @@
  # CHANGES to the 'XMod' package
 
-## 2.92 -> 2.92dev (17/12/2024)
+## 2.92 -> 2.92dev (03/01/2025)
+ * (20/12/24) changed the examples of induced crossed modules in section 7.2
  * (17/12/24) implemented (Inner)ActorCat1Group, fixing issue #144
  * (20/11/24) added AutomorphismPermGroup method for cat1-groups
               renamed PermAutomorphismAsXModMorphism as

diff --git a/PackageInfo.g b/PackageInfo.g
@@ -8,7 +8,7 @@ SetPackageInfo( rec(
 PackageName := "XMod",
 Subtitle := "Crossed Modules and Cat1-Groups",
 Version := "2.92dev",
-Date := "17/12/2024", # dd/mm/yyyy format
+Date := "03/01/2025", # dd/mm/yyyy format
 License := "GPL-2.0-or-later",
 
 Persons := [
@@ -127,7 +127,7 @@ Keywords := ["crossed module", "cat1-group", "derivation", "section",
 AutoDoc := rec(
     TitlePage := rec(
         Copyright := Concatenation(
-            "&copyright; 1996-2024, Chris Wensley et al. <P/>\n", 
+            "© 1996-2025, Chris Wensley et al. <P/>\n", 
             "The &XMod; package is free software; you can redistribute it ", 
             "and/or modify it under the terms of the GNU General ", 
             "Public License as published by the Free Software Foundation; ", 

diff --git a/README.md b/README.md
@@ -20,7 +20,7 @@ Functions for crossed squares and cat2-groups have been added during 2019/20.
 
 ## Copyright
 
-The 'XMod' package is Copyright {\copyright} Chris Wensley et al, 1997--2024. 
+The 'XMod' package is Copyright © Chris Wensley et al, 1997--2025. 
 
 'XMod' is free software; you can redistribute it and/or modify
 it under the terms of the GNU General Public License as published by
@@ -45,16 +45,15 @@ Once these prerequisites are in place, proceed as follows:
 
  * Unpack `xmod-<version_number>.tar.gz` in the `pkg` subdirectory of the GAP root directory.
  * From within GAP load the package with:
-
+    ```
     gap> LoadPackage( "xmod" );
-
     true
-
- * The file manual.pdf is in the `doc' subdirectory.
+    ```
+ * The file manual.pdf is in the `doc` subdirectory.
  * To run the test file read `testall.g` from the `tst` subdirectory. 
 
 ## Contact
 
 If you have a question relating to 'XMod', encounter any problems, or have a suggestion for extending the package in any way, please 
- * email: `[email protected]` 
+ * email: <mailto:[email protected]> 
  * or report an issue at: <https://github.com/gap-packages/xmod/issues/new> 
diff --git a/doc/gp2ind.xml b/doc/gp2ind.xml
@@ -2,7 +2,7 @@
 <!--                                                                     -->
 <!--  gp2ind.xml            XMod documentation            Chris Wensley  -->
 <!--                                                        & Murat Alp  -->
-<!--  Copyright (C) 2001-2024, Chris Wensley et al,                      --> 
+<!--  Copyright (C) 2001-2025, Chris Wensley et al,                      --> 
 <!--  School of Computer Science, Bangor University, U.K.                --> 
 <!--                                                                     -->
 <!-- ------------------------------------------------------------------- -->
@@ -166,81 +166,121 @@ These constructions use Tietze transformation routines in
 the library file <C>tietze.gi</C>.
 <P/>
 As a first, surjective example, we take for <M>\calX</M> 
-the normal inclusion crossed module of <C>a4</C> in <C>s4</C>,
-and for <M>\iota</M> the surjection from <C>s4</C> to <C>s3</C> 
-with kernel <C>k4</C>.  
-The induced crossed module is isomorphic to <C>X3 = [c3->s3]</C>.
+a central extension crossed module of dihedral groups, 
+<M>(d_{24} \to d_{12})</M>,
+and for <M>\iota</M> a surjection <M>d_{12} \to s_3</M> 
+with kernel <M>c_2</M>.  
+The induced crossed module is isomorphic to <M>(d_{12} \to s_3)</M>.
 </Description>
 </ManSection>
 <P/>
 <Example>
 <![CDATA[
-gap> s4gens := GeneratorsOfGroup( s4 );
-[ (1,2), (2,3), (3,4) ]
-gap> a4gens := GeneratorsOfGroup( a4 );
-[ (1,2,3), (2,3,4) ]
-gap> s3b := Group( (5,6),(6,7) );;  SetName( s3b, "s3b" );
-gap> epi := GroupHomomorphismByImages( s4, s3b, s4gens, [(5,6),(6,7),(5,6)] );;
-gap> X4 := XModByNormalSubgroup( s4, a4 );;
-gap> indX4 := InducedXModBySurjection( X4, epi );
-[a4/ker->s3b]
-gap> Display( indX4 );
-
-Crossed module [a4/ker->s3b] :- 
-: Source group a4/ker has generators:
-  [ (1,3,2), (1,2,3) ]
-: Range group s3b has generators:
-  [ (5,6), (6,7) ]
+gap> a := (6,7,8,9)(10,11,12);;  b := (7,9)(11,12);;
+gap> d24 := Group( [ a, b ] );;
+gap> SetName( d24, "d24" );
+gap> c := (1,2)(3,4,5);;  d := (4,5);;
+gap> d12 := Group( [ c, d ] );;
+gap> SetName( d12, "d12" );
+gap> bdy := GroupHomomorphismByImages( d24, d12, [a,b], [c,d] );;
+gap> X24 := XModByCentralExtension( bdy );
+[d24->d12]
+gap> e := (13,14,15);;  f := (14,15);;
+gap> s3 := Group( [ e, f ] );;
+gap> SetName( s3, "s3" );;
+gap> epi := GroupHomomorphismByImages( d12, s3, [c,d], [e,f] );;
+gap> iX24 := InducedXModBySurjection( X24, epi );
+[d24/ker->s3]
+gap> Display( iX24 );               
+Crossed module [d24/ker->s3] :- 
+: Source group d24/ker has generators:
+  [ ( 1,11, 5, 4,10, 8)( 2,12, 6, 3, 9, 7), 
+  ( 1, 2)( 3, 4)( 5, 9)( 6,10)( 7,11)( 8,12) ]
+: Range group s3 has generators:
+  [ (13,14,15), (14,15) ]
 : Boundary homomorphism maps source generators to:
-  [ (5,6,7), (5,7,6) ]
+  [ (13,14,15), (14,15) ]
 : Action homomorphism maps range generators to automorphisms:
-  (5,6) --> { source gens --> [ (1,2,3), (1,3,2) ] }
-  (6,7) --> { source gens --> [ (1,2,3), (1,3,2) ] }
+  (13,14,15) --> { source gens --> [ ( 1,11, 5, 4,10, 8)( 2,12, 6, 3, 9, 7), 
+  ( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,10)(11,12) ] }
+  (14,15) --> { source gens --> [ ( 1, 8,10, 4, 5,11)( 2, 7, 9, 3, 6,12), 
+  ( 1, 2)( 3, 4)( 5, 9)( 6,10)( 7,11)( 8,12) ] }
   These 2 automorphisms generate the group of automorphisms.
-
-gap> morX4 := MorphismOfInducedXMod( indX4 );
-[[a4->s4] => [a4/ker->s3b]]
+gap> morX24 := MorphismOfInducedXMod( iX24 );
+[[d24->d12] => [d24/ker->s3]]
 ]]>
 </Example>
 
 For a second, injective example we take for <M>\calX</M> 
-the automorphism crossed module <C>XAq8</C> of <Ref Oper="CoproductXMod"/>, 
-and for <M>\iota</M> an inclusion of <C>s4b</C> in <C>s5</C>. 
-The resulting source group is <C>SL(2,5)</C>. 
+the result <C>iX24</C> of the previous example
+and for <M>\iota</M> an inclusion of <M>s_3</M> in <M>s_4</M>. 
+The resulting source group has size <M>96</M>. 
 <P/>
 <Example>
 <![CDATA[
-gap> iso4 := IsomorphismGroups( s4b, s4 );; 
-gap> s5 := Group( (1,2,3,4,5), (4,5) );; 
-gap> SetName( s5, "s5" ); 
-gap> inc45 := InclusionMappingGroups( s5, s4 );;
-gap> iota45 := iso4 * inc45;; 
-gap> indXAq8 := InducedXMod( XAq8, iota45 );
-i*(XAq8)
-gap> Size2d( indXAq8 );
-[ 120, 120 ]
-gap> StructureDescription( indXAq8 );          
-[ "SL(2,5)", "S5" ]
+gap> g := (16,17,18);;  h := (16,17,18,19);;
+gap> s4 := Group( [ g, h ] );;
+gap> SetName( s4, "s4" );;
+gap> iota := GroupHomomorphismByImages( s3, s4, [e,f], [g^2*h^2,g*h^-1] );
+[ (13,14,15), (14,15) ] -> [ (17,18,19), (18,19) ]
+gap> iiX24 := InducedXModByCopower( iX24, iota, [ ] );
+i*([d24/ker->s3])
+gap> Size2d( iiX24 );               
+[ 96, 24 ]
+gap> StructureDescription( iiX24 );
+[ "C2 x GL(2,3)", "S4" ]
 ]]>
 </Example>
 
-For a third example we use the version <C>InducedXMod(Q,R,S)</C> 
-of this global function, with <M>Q \geqslant R \unrhd S</M>. 
-We take the identity mapping on <C>s3c</C> as boundary,
-and the inclusion of <C>s3c</C> in <C>s4</C> as <M>\iota</M>.
-The induced group is a general linear group <C>GL(2,3)</C>.
+For a third example we combine the previous two examples by 
+taking for <M>\iota</M> the more general case <C>alpha = theta*iota</C>.
+The resulting <C>jX24</C> is isomorphic to, 
+but not identical to, <C>iiX24</C>.
 <P/>
 <Example>
 <![CDATA[
-gap> s3c := Subgroup( s4, [ (2,3), (3,4) ] );;  
-gap> SetName( s3c, "s3c" );
-gap> indXs3c := InducedXMod( s4, s3c, s3c );
-i*([s3c->s3c])
-gap> StructureDescription( indXs3c );
-[ "GL(2,3)", "S4" ]
+gap> alpha := CompositionMapping( iota, epi );
+[ (1,2)(3,4,5), (4,5) ] -> [ (17,18,19), (18,19) ]
+gap> jX24 := InducedXMod( X24, alpha );;
+gap> StructureDescription( jX24 );
+[ "C2 x GL(2,3)", "S4" ]
 ]]>
 </Example>
 
+For a fourth example we use the version <C>InducedXMod(Q,R,S)</C> 
+of this global function, with a normal inclusion crossed module 
+<M>(S \to R)</M> and an inclusion mapping <M>R \to Q</M>.
+We take <M>(c_6 \to d_{12})</M> as <M>\calX</M> 
+and the inclusion of <M>d_{12}</M> in <M>d_{24}</M> as <M>\iota</M>.
+<P/>
+<Example>
+<![CDATA[
+## Section 7.2.1 : Example 4 
+gap> d12b := Subgroup( d24, [ a^2, b ] );;
+gap> SetName( d12b, "d12b" ); 
+gap> c6b := Subgroup( d12b, [ a^2 ] );; 
+gap> SetName( c6b, "c6b" );  
+gap> X12 := InducedXMod( d24, d12b, c6b );
+i*([c6b->d12b])
+gap> StructureDescription( X12 );
+[ "C6 x C6", "D24" ]
+gap> Display( MorphismOfInducedXMod( X12 ) );
+Morphism of crossed modules :- 
+: Source = [c6b->d12b] with generating sets:
+  [ ( 6, 8)( 7, 9)(10,12,11) ]
+  [ ( 6, 8)( 7, 9)(10,12,11), ( 7, 9)(11,12) ]
+:  Range = i*([c6b->d12b]) with generating sets:
+  [ ( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15), 
+  ( 4, 6, 8)( 5, 7, 9)(10,12,14)(11,13,15), 
+  ( 4,10)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15), (1,2,3) ]
+  [ ( 6, 7, 8, 9)(10,11,12), ( 7, 9)(11,12) ]
+: Source Homomorphism maps source generators to:
+  [ ( 4, 9, 6, 5, 8, 7)(10,15,12,11,14,13) ]
+: Range Homomorphism maps range generators to:
+  [ ( 6, 8)( 7, 9)(10,12,11), ( 7, 9)(11,12) ]
+#]]>
+</Example>
+
 <ManSection>
    <Oper Name="AllInducedXMods"
          Arg="Q" />
@@ -254,13 +294,11 @@ and <C>S</C> runs over all non-trivial normal subgroups of <C>R</C>.
 <Example>
 <![CDATA[
 gap> all := AllInducedXMods( q8 );;
-gap> ids := List( all, x -> IdGroup(x) );;
-gap> Sort( ids );
-gap> ids;
-[ [ [ 1, 1 ], [ 8, 4 ] ], [ [ 1, 1 ], [ 8, 4 ] ], [ [ 1, 1 ], [ 8, 4 ] ], 
-  [ [ 1, 1 ], [ 8, 4 ] ], [ [ 4, 2 ], [ 8, 4 ] ], [ [ 4, 2 ], [ 8, 4 ] ], 
-  [ [ 4, 2 ], [ 8, 4 ] ], [ [ 16, 2 ], [ 8, 4 ] ], [ [ 16, 2 ], [ 8, 4 ] ], 
-  [ [ 16, 2 ], [ 8, 4 ] ], [ [ 16, 14 ], [ 8, 4 ] ] ]
+gap> L := List( all, x -> Source( x ) );;
+gap> Sort( L, function(g,h) return Size(g) < Size(h); end );;
+gap> List( L, x -> StructureDescription( x ) );
+[ "1", "1", "1", "1", "C2 x C2", "C2 x C2", "C2 x C2", "C4 x C4", "C4 x C4", 
+  "C4 x C4", "C2 x C2 x C2 x C2" ]
 ]]>
 </Example>
 

diff --git a/doc/gp3xsq.xml b/doc/gp3xsq.xml
@@ -2,7 +2,7 @@
 <!--                                                                     -->
 <!--  gp3xsq.xml            XMod documentation            Chris Wensley  -->
 <!--                                                                     -->
-<!--  Copyright (C) 1996-2022, Chris Wensley et al,                      --> 
+<!--  Copyright (C) 1996-2024, Chris Wensley et al,                      --> 
 <!--  School of Computer Science, Bangor University, U.K.                --> 
 <!--                                                                     -->
 <!-- ------------------------------------------------------------------- -->
@@ -360,7 +360,12 @@ The crossed pairing is given by
 <Display>
 \boxtimes \;:\;  R \times W \,\to\, S, \quad (r,\chi) \,\mapsto\, \chi r~.
 </Display>
-This is implemented as <C>ActorCrossedSquare(X0);</C>.
+This is implemented as <C>ActorCrossedSquare(X0)</C>.
+<P/>
+The example constructs <C>XSact</C>, the actor crossed square 
+of the crossed module <C>X20</C>. 
+This crossed square is converted to a cat<M>^2</M>-group <C>C2act</C> 
+in section <Ref Sect="cat2-xsq"/>.
 </Description>
 </ManSection>
 <P/>

diff --git a/lib/gp2ind.gd b/lib/gp2ind.gd
@@ -1,8 +1,8 @@
-###############################################################################
+##############################################################################
 ##
-#W  gp2ind.gd                   GAP4 package `XMod'               Chris Wensley
+#W  gp2ind.gd                   GAP4 package `XMod'             Chris Wensley
 ##
-#Y  Copyright (C) 2001-2020, Chris Wensley et al,  
+#Y  Copyright (C) 2001-2024, Chris Wensley et al,  
 #Y  School of Computer Science, Bangor University, U.K. 
 ##  
 ##  This file declares functions for induced crossed modules. 
@@ -37,7 +37,7 @@
 
 #############################################################################
 ##
-##  #A  InducedXModData( <IX> )
+#A  InducedXModData( <IX> )
 ##
 ##  DeclareAttribute( "InducedXModData", Is2DimensionalDomain, "mutable" );
 
@@ -73,17 +73,20 @@ DeclareAttribute( "MorphismOfInducedXMod", IsInducedXMod );
 #O  InducedXModFromTrivialRange( <xmod>, <hom> )
 ##
 DeclareGlobalFunction( "InducedXMod" );
-DeclareOperation( "InducedXModBySurjection", [ IsXMod, IsGroupHomomorphism ] );
-DeclareOperation( "InducedXModByCoproduct", [ IsXMod, IsGroupHomomorphism ] );
-DeclareOperation( "InducedXModByBijection", [ IsXMod, IsGroupHomomorphism ] );
+DeclareOperation( "InducedXModBySurjection", 
+    [ IsXMod, IsGroupHomomorphism ] );
+DeclareOperation( "InducedXModByCoproduct", 
+    [ IsXMod, IsGroupHomomorphism ] );
+DeclareOperation( "InducedXModByBijection", 
+    [ IsXMod, IsGroupHomomorphism ] );
 DeclareOperation( "InducedXModByCopower", 
     [ IsXMod, IsGroupHomomorphism, IsList ] );
 DeclareOperation( "InducedXModFromTrivialSource", 
     [ IsXMod, IsGroupHomomorphism ] );
 DeclareOperation( "InducedXModFromTrivialRange", 
     [ IsXMod, IsGroupHomomorphism ] );
 
-##############################################################################
+#############################################################################
 ##
 #F  InducedCat1Group( <args> )
 #O  InducedCat1Data( <grp>, <hom>, <trans> )
@@ -94,7 +97,7 @@ DeclareOperation( "InducedCat1Data",
     [ IsCat1Group, IsGroupHomomorphism, IsList ] );
 DeclareOperation( "InducedCat1GroupByFreeProduct", [ IsList ] );
 
-##############################################################################
+#############################################################################
 ##
 #O  AllInducedXMods( <grp> )
 #O  AllInducedCat1Groups( <grp> )