Derivations and Sections
up 2d-mapping of 2d-object
Whitehead Multiplication
Whitehead multiplication
derivation, of crossed module
Whitehead monoid
The Whitehead monoid {\rm Der}(\mathcal{X}) of \mathcal{X} was defined in to be the monoid of all derivations from R to S, that is the set of all maps \chi : R \to S, with Whitehead multiplication \star (on the right) satisfying: {\bf Der\ 1}: \chi(qr) ~=~ (\chi q)^{r} \; (\chi r), \qquad {\bf Der\ 2}: (\chi_1 \star \chi_2)(r) ~=~ (\chi_2 r)(\chi_1 r)(\chi_2 \partial \chi_1 r). \end{eqnarray*} The zero map is the identity for this composition. regular derivation Invertible elements in the monoid are called regular. Whitehead group The Whitehead group of \mathcal{X} is the group of regular derivations in {\rm Der}(\mathcal{X} ). In the next chapter the actor of \mathcal{X} is defined as a crossed module whose source and range are permutation representations of the Whitehead group and the automorphism group of \mathcal{X}.

section, of cat1-group
The construction for cat1-groups equivalent to the derivation of a crossed module is the section. Whitehead multiplication The monoid of sections of \mathcal{C} = (e;t,h : G \to R) is the set of group homomorphisms \xi : R \to G, with Whitehead multiplication \star, (on the right) satisfying: {\bf Sect\ 1}: t \xi ~=~ {\rm id}_R, \quad {\bf Sect\ 2}: (\xi_1 \star \xi_2)(r) ~=~ (\xi_1 r)(e h \xi_1 r)^{-1}(\xi_2 h \xi_1 r) ~=~ (\xi_2 h \xi_1 r)(e h \xi_1 r)^{-1}(\xi_1 r). The embedding e is the identity for this composition, and h(\xi_1 \star \xi_2) = (h \xi_1)(h \xi_2). A section is regular when h \xi is an automorphism, and the group of regular sections is isomorphic to the Whitehead group.

If \epsilon denotes the inclusion of S = {\rm ker} t in G then \partial = h \epsilon : S \to R and \xi r ~=~ (e r)(e \chi r) ~=~ (r, \chi r) determines a section \xi of \mathcal{C} in terms of the corresponding derivation \chi of \mathcal{X}, and conversely.
Derivations are stored like group homomorphisms by specifying the images of a generating set. Images of the remaining elements may then be obtained using axiom {\bf Der\ 1}. The function IsDerivation is automatically called to check that this procedure is well-defined.

In the following example a cat1-group C3 and the associated crossed module X3 are constructed, where X3 is isomorphic to the inclusion of the normal cyclic group c3 in the symmetric group s3.
gap> g18 := Group( (1,2,3), (4,5,6), (2,3)(5,6) );; gap> SetName( g18, "g18" ); gap> gen18 := GeneratorsOfGroup( g18 );; gap> g1 := gen18[1];; g2 := gen18[2];; g3 := gen18[3];; gap> s3 := Subgroup( g18, gen18{[2..3]} );; gap> SetName( s3, "s3" );; gap> t := GroupHomomorphismByImages( g18, s3, gen18, [g2,g2,g3] );; gap> h := GroupHomomorphismByImages( g18, s3, gen18, [(),g2,g3] );; gap> e := GroupHomomorphismByImages( s3, g18, [g2,g3], [g2,g3] );; gap> C3 := Cat1( t, h, e ); [g18=>s3] gap> SetName( Kernel(t), "c3" );; gap> X3 := XModOfCat1( C3 );; gap> Display( X3 ); Crossed module [c3->s3] :- : Source group has generators: [ ( 1, 2, 3)( 4, 6, 5) ] : Range group has generators: [ (4,5,6), (2,3)(5,6) ] : Boundary homomorphism maps source generators to: [ (4,6,5) ] : Action homomorphism maps range generators to automorphisms: (4,5,6) --> { source gens --> [ (1,2,3)(4,6,5) ] } (2,3)(5,6) --> { source gens --> [ (1,3,2)(4,5,6) ] } These 2 automorphisms generate the group of automorphisms. : associated cat1-group is [g18=>s3]

gap> imchi := [ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ];; gap> chi := DerivationByImages( X3, imchi ); DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ] )
Sections are group homomorphisms, so do not need a special representation. Operations SectionByDerivation and DerivationBySection convert derivations to sections, and vice-versa, calling Cat1OfXMod and XModOfCat1 automatically.

Two strategies for calculating derivations and sections are implemented, see . The default method for AllDerivations is to search for all possible sets of images using a backtracking procedure, and when all the derivations are found it is not known which are regular. In the &GAP;3 version of this package, the default method for AllSections( <C> ) was to compute all endomorphisms on the range group R of C as possibilities for the composite h \xi. A backtrack method then found possible images for such a section. In the current version the derivations of the associated crossed module are calculated, and these are all converted to sections using SectionByDerivation.
gap> xi := SectionByDerivation( chi ); [ (4,5,6), (2,3)(5,6) ] -> [ (1,2,3), (1,2)(4,6) ]
Whitehead Groups and Monoids
There are two functions to determine the elements of the Whitehead group and the Whitehead monoid of \mathcal{X}0, namely RegularDerivations and AllDerivations. (The functions RegularSections and AllSections perform corresponding tasks for a cat1-group.)

Using our example X3 we find that there are just nine derivations, six of them regular, and the associated group is isomorphic to s3.
gap> all3 := AllDerivations( X3 );; gap> imall3 := ImagesList( all3 );; Display( imall3 ); [ [ (), () ], [ (), ( 1, 2, 3)( 4, 6, 5) ], [ (), ( 1, 3, 2)( 4, 5, 6) ], [ ( 1, 2, 3)( 4, 6, 5), () ], [ ( 1, 2, 3)( 4, 6, 5), ( 1, 2, 3)( 4, 6, 5) ], [ ( 1, 2, 3)( 4, 6, 5), ( 1, 3, 2)( 4, 5, 6) ], [ ( 1, 3, 2)( 4, 5, 6), () ], [ ( 1, 3, 2)( 4, 5, 6), ( 1, 2, 3)( 4, 6, 5) ], [ ( 1, 3, 2)( 4, 5, 6), ( 1, 3, 2)( 4, 5, 6) ] ] gap> KnownAttributesOfObject( all3 ); [ "Object2d", "ImagesList", "AllOrRegular", "ImagesTable" ] gap> Display( ImagesTable( all3 ) ); [ [ 1, 1, 1, 1, 1, 1 ], [ 1, 1, 1, 2, 2, 2 ], [ 1, 1, 1, 3, 3, 3 ], [ 1, 2, 3, 1, 2, 3 ], [ 1, 2, 3, 2, 3, 1 ], [ 1, 2, 3, 3, 1, 2 ], [ 1, 3, 2, 1, 3, 2 ], [ 1, 3, 2, 2, 1, 3 ], [ 1, 3, 2, 3, 2, 1 ] ]
The Whitehead product \chi_1 \star \chi_2 is implemented as CompositeDerivation( <chi1>, <chi2> ). The composite of two sections is just the composite of homomorphisms.
gap> reg3 := RegularDerivations( X3 );; gap> imder3 := ImagesList( reg3 );; gap> chi4 := DerivationByImages( X3, imder3[4] ); DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ ( 1, 3, 2)( 4, 5, 6), () ] ) gap> chi5 := DerivationByImages( X3, imder3[5] ); DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ ( 1, 3, 2)( 4, 5, 6), ( 1, 2, 3)( 4, 6, 5) ] ) gap> im4 := ImagePositions( chi4 ); [ 1, 3, 2, 1, 3, 2 ] gap> im5 := ImagePositions( chi5 ); [ 1, 3, 2, 2, 1, 3 ] gap> chi45 := chi4 * chi5; DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (), ( 1, 2, 3)( 4, 6, 5) ] ) gap> im45 := ImagePositions( chi45 ); [ 1, 1, 1, 2, 2, 2 ] gap> pos := Position( imder3, GeneratorImages( chi45 ) ); 2
Multiplication tables for the Whitehead group or monoid enable the construction of permutation or transformation representations.
gap> wgt3 := WhiteheadGroupTable( X3 );; Display( wgt3 ); [ [ 1, 2, 3, 4, 5, 6 ], [ 2, 3, 1, 5, 6, 4 ], [ 3, 1, 2, 6, 4, 5 ], [ 4, 6, 5, 1, 3, 2 ], [ 5, 4, 6, 2, 1, 3 ], [ 6, 5, 4, 3, 2, 1 ] ] gap> wpg3 := WhiteheadPermGroup( X3 ); Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ]) gap> wmt3 := WhiteheadMonoidTable( X3 );; Display( wmt3 ); [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ], [ 2, 3, 1, 5, 6, 4, 8, 9, 7 ], [ 3, 1, 2, 6, 4, 5, 9, 7, 8 ], [ 4, 4, 4, 4, 4, 4, 4, 4, 4 ], [ 5, 5, 5, 5, 5, 5, 5, 5, 5 ], [ 6, 6, 6, 6, 6, 6, 6, 6, 6 ], [ 7, 9, 8, 4, 6, 5, 1, 3, 2 ], [ 8, 7, 9, 5, 4, 6, 2, 1, 3 ], [ 9, 8, 7, 6, 5, 4, 3, 2, 1 ] ] gap> wtm3 := WhiteheadTransMonoid( X3 ); Monoid( [ Transformation( [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] ), Transformation( [ 2, 3, 1, 5, 6, 4, 8, 9, 7 ] ), Transformation( [ 3, 1, 2, 6, 4, 5, 9, 7, 8 ] ), Transformation( [ 4, 4, 4, 4, 4, 4, 4, 4, 4 ] ), Transformation( [ 5, 5, 5, 5, 5, 5, 5, 5, 5 ] ), Transformation( [ 6, 6, 6, 6, 6, 6, 6, 6, 6 ] ), Transformation( [ 7, 9, 8, 4, 6, 5, 1, 3, 2 ] ), Transformation( [ 8, 7, 9, 5, 4, 6, 2, 1, 3 ] ), Transformation( [ 9, 8, 7, 6, 5, 4, 3, 2, 1 ] ) ], ... )