  
  [1X5 [33X[0;0YWhitehead group of a crossed module[133X[101X
  
  
  [1X5.1 [33X[0;0YDerivations and Sections[133X[101X
  
  [33X[0;0YThe  Whitehead  monoid  [22XDer(calX)[122X  of  [22XcalX[122X was defined in [Whi48] to be the
  monoid  of all [13Xderivations[113X from [22XR[122X to [22XS[122X, that is the set of all maps [22Xχ : R ->
  S[122X, with [13XWhitehead product[113X [22X⋆[122X (on the [13Xright[113X) satisfying:[133X
  
  
  [24X[33X[0;6Y{\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).[133X
  
  [124X
  
  [33X[0;0YIt  easily follows that [22Xχ 1 = 1[122X and [22Xχ(r^-1) = ((χ r)^-1)^r^-1}[122X, and that the
  zero  map  is  the identity for this composition. Invertible elements in the
  monoid  are  called  [13Xregular[113X.  The  Whitehead  group of [22XcalX[122X is the group of
  regular  derivations in [22XDer(calX )[122X. In the next chapter the [13Xactor[113X of [22XcalX[122X is
  defined  as  a  crossed  module  whose  source  and  range  are  permutation
  representations of the Whitehead group and the automorphism group of [22XcalX[122X.[133X
  
  [33X[0;0YThe  construction for cat[22X^1[122X-groups equivalent to the derivation of a crossed
  module  is the [13Xsection[113X. The monoid of sections of [22XcalC = (e;t,h : G -> R)[122X is
  the  set  of group homomorphisms [22Xξ : R -> G[122X, with Whitehead multiplication [22X⋆[122X
  (on the [13Xright[113X) satisfying:[133X
  
  
  [24X[33X[0;6Y{\bf Sect\ 1}: t \circ \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).[133X
  
  [124X
  
  [33X[0;0YThe  embedding [22Xe[122X is the identity for this composition, and [22Xh(ξ_1 ⋆ ξ_2) = (h
  ξ_1)(h ξ_2)[122X. A section is [13Xregular[113X when [22Xh ξ[122X is an automorphism, and the group
  of regular sections is isomorphic to the Whitehead group.[133X
  
  [33X[0;0YIf [22Xϵ[122X denotes the inclusion of [22XS = ker t[122X in [22XG[122X then [22X∂ = h ϵ : S -> R[122X and[133X
  
  
  [24X[33X[0;6Y\xi  r ~=~ (e r)(\epsilon \chi r), \quad\mbox{which equals}\quad (r, \chi r)
  ~\in~ R \ltimes S,[133X
  
  [124X
  
  [33X[0;0Ydetermines a section [22Xξ[122X of [22XcalC[122X in terms of the corresponding derivation [22Xχ[122X of
  [22XcalX[122X, and conversely.[133X
  
  [1X5.1-1 DerivationByImages[101X
  
  [33X[1;0Y[29X[2XDerivationByImages[102X( [3XX0[103X, [3Xims[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsDerivation[102X( [3Xmap[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsUp2DimensionalMapping[102X( [3Xchi[103X ) [32X property[133X
  [33X[1;0Y[29X[2XUpGeneratorImages[102X( [3Xchi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XUpImagePositions[102X( [3Xchi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XObject2d[102X( [3Xchi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDerivationImage[102X( [3Xchi[103X, [3Xr[103X ) [32X operation[133X
  
  [33X[0;0YA derivation [22Xχ[122X is stored like a group homomorphisms by specifying the images
  of the generating set [10XStrongGeneratorsStabChain( StabChain(R) )[110X of the range
  [22XR[122X. This set of images is stored as the attribute [10XUpGeneratorImages[110X of [22Xχ[122X. The
  function  [10XIsDerivation[110X  is automatically called to check that this procedure
  is  well-defined.  The  attribute [10XObject2d[110X[22X(χ)[122X returns the underlying crossed
  module.[133X
  
  [33X[0;0YImages  of  the  remaining  elements  may  be  obtained  using  axiom [22XDer 1[122X.
  [10XUpImagePositions(chi)[110X  is  the list of the images under [22Xχ[122X of [10XElements(R)[110X and
  [10XDerivationImage(chi,r)[110X returns [22Xχ r[122X.[133X
  
  [33X[0;0YIn  the following example a cat[22X^1[122X-group [10XC3[110X and the associated crossed module
  [10XX3[110X  are  constructed,  where [10XX3[110X is isomorphic to the inclusion of the normal
  cyclic group [10Xc3[110X in the symmetric group [10Xs3[110X. The derivation [22Xχ_1[122X maps [10Xc3[110X to the
  identity and the other [22X3[122X elements to [22X(1,2,3)(4,6,5)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xg18 := Group( (1,2,3), (4,5,6), (2,3)(5,6) );;[127X[104X
    [4X[25Xgap>[125X [27XSetName( g18, "g18" );[127X[104X
    [4X[25Xgap>[125X [27Xgen18 := GeneratorsOfGroup( g18 );;[127X[104X
    [4X[25Xgap>[125X [27Xg1 := gen18[1];;  g2 := gen18[2];;  g3 := gen18[3];;[127X[104X
    [4X[25Xgap>[125X [27Xs3 := Subgroup( g18, gen18{[2..3]} );;[127X[104X
    [4X[25Xgap>[125X [27XSetName( s3, "s3" );;[127X[104X
    [4X[25Xgap>[125X [27Xt := GroupHomomorphismByImages( g18, s3, gen18, [g2,g2,g3] );;[127X[104X
    [4X[25Xgap>[125X [27Xh := GroupHomomorphismByImages( g18, s3, gen18, [(),g2,g3] );;[127X[104X
    [4X[25Xgap>[125X [27Xe := GroupHomomorphismByImages( s3, g18, [g2,g3], [g2,g3] );;[127X[104X
    [4X[25Xgap>[125X [27XC3 := Cat1Group( t, h, e );[127X[104X
    [4X[28X[g18=>s3][128X[104X
    [4X[25Xgap>[125X [27XSetName( Kernel(t), "c3" );;[127X[104X
    [4X[25Xgap>[125X [27XX3 := XModOfCat1Group( C3 );[127X[104X
    [4X[28X[c3->s3][128X[104X
    [4X[25Xgap>[125X [27XR3 := Range( X3 );;[127X[104X
    [4X[25Xgap>[125X [27XStrongGeneratorsStabChain( StabChain( R3 ) );[127X[104X
    [4X[28X[ (4,5,6), (2,3)(5,6) ][128X[104X
    [4X[25Xgap>[125X [27Xchi1 := DerivationByImages( X3, [ (), (1,2,3)(4,6,5) ] );[127X[104X
    [4X[28XDerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [128X[104X
    [4X[28X[ (), (1,2,3)(4,6,5) ] )[128X[104X
    [4X[25Xgap>[125X [27X[ IsUp2DimensionalMapping( chi1 ), IsDerivation( chi1 ) ];[127X[104X
    [4X[28X[ true, true ][128X[104X
    [4X[25Xgap>[125X [27XObject2d( chi1 );[127X[104X
    [4X[28X[c3->s3][128X[104X
    [4X[25Xgap>[125X [27XUpGeneratorImages( chi1 ); [127X[104X
    [4X[28X[ (), (1,2,3)(4,6,5) ][128X[104X
    [4X[25Xgap>[125X [27XUpImagePositions( chi1 );[127X[104X
    [4X[28X[ 1, 1, 1, 2, 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XDerivationImage( chi1, (2,3)(4,5) );[127X[104X
    [4X[28X(1,2,3)(4,6,5)[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.1-2 PrincipalDerivation[101X
  
  [33X[1;0Y[29X[2XPrincipalDerivation[102X( [3XX0[103X, [3Xs[103X ) [32X operation[133X
  
  [33X[0;0YThe [13Xprincipal derivation[113X determined by [22Xs ∈ S[122X is the derivation [22Xη_s : R -> S,
  r ↦ (s^-1)^rs[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xeta := PrincipalDerivation( X3, (1,2,3)(4,6,5) );[127X[104X
    [4X[28XDerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (), (1,3,2)(4,5,6) ] )[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.1-3 SectionByHomomorphism[101X
  
  [33X[1;0Y[29X[2XSectionByHomomorphism[102X( [3XC[103X, [3Xhom[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XIsSection[102X( [3Xxi[103X ) [32X property[133X
  [33X[1;0Y[29X[2XUpHomomorphism[102X( [3Xxi[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XSectionByDerivation[102X( [3Xchi[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XDerivationBySection[102X( [3Xxi[103X ) [32X operation[133X
  
  [33X[0;0YSections [13Xare[113X group homomorphisms but, although they do not require a special
  representation,  one  is  provided  in  the category [10XIsUp2DimensionalMapping[110X
  having   attributes   [10XObject2d[110X,   [10XUpHomomorphism[110X   and   [10XUpGeneratorsImages[110X.
  Operations  [10XSectionByDerivation[110X  and [10XDerivationBySection[110X convert derivations
  to   sections,   and   vice-versa,   calling   [2XCat1GroupOfXMod[102X  ([14X2.5-3[114X)  and
  [2XXModOfCat1Group[102X ([14X2.5-3[114X) automatically.[133X
  
  [33X[0;0YTwo strategies for calculating derivations and sections are implemented, see
  [AW00].  The  default method for [2XAllDerivations[102X ([14X5.2-1[114X) 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 early versions
  of  this  package,  the default method for [10XAllSections( <C> )[110X was to compute
  all  endomorphisms  on  the  range  group  [10XR[110X  of  [10XC[110X as possibilities for the
  composite  [22Xh  ξ[122X.  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
  [2XSectionByDerivation[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xhom2 := GroupHomomorphismByImages( s3, g18, [ (4,5,6), (2,3)(5,6) ], [127X[104X
    [4X[25X>[125X [27X[ (1,3,2)(4,6,5), (1,2)(4,6) ] );;[127X[104X
    [4X[25Xgap>[125X [27Xxi2 := SectionByHomomorphism( C3, hom2 );                                 [127X[104X
    [4X[28XSectionByHomomorphism( s3, g18, [ (4,5,6), (2,3)(5,6) ], [128X[104X
    [4X[28X[ (1,3,2)(4,6,5), (1,2)(4,6) ] )[128X[104X
    [4X[25Xgap>[125X [27X[ IsUp2DimensionalMapping( xi2 ), IsSection( xi2 ) ];[127X[104X
    [4X[28X[ true, true ][128X[104X
    [4X[25Xgap>[125X [27XObject2d( xi2 );[127X[104X
    [4X[28X[g18 => s3][128X[104X
    [4X[25Xgap>[125X [27XUpHomomorphism( xi2 );         [127X[104X
    [4X[28X[ (4,5,6), (2,3)(5,6) ] -> [ (1,3,2)(4,6,5), (1,2)(4,6) ][128X[104X
    [4X[25Xgap>[125X [27XUpGeneratorImages( xi2 );[127X[104X
    [4X[28X[ (1,3,2)(4,6,5), (1,2)(4,6) ][128X[104X
    [4X[25Xgap>[125X [27Xchi2 := DerivationBySection( xi2 );[127X[104X
    [4X[28XDerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [128X[104X
    [4X[28X[ (1,3,2)(4,5,6), (1,2,3)(4,6,5) ] )[128X[104X
    [4X[25Xgap>[125X [27Xxi1 := SectionByDerivation( chi1 );[127X[104X
    [4X[28XSectionByHomomorphism( s3, g18, [ (4,5,6), (2,3)(5,6) ], [128X[104X
    [4X[28X[ (1,2,3), (1,2)(4,6) ] )[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.1-4 IdentityDerivation[101X
  
  [33X[1;0Y[29X[2XIdentityDerivation[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIdentitySection[102X( [3XC0[103X ) [32X attribute[133X
  
  [33X[0;0YThe identity derivation maps the range group to the identity subgroup of the
  source, while the identity section is just the range embedding considered as
  a section.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XIdentityDerivation( X3 ); [127X[104X
    [4X[28XDerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (), () ] )[128X[104X
    [4X[25Xgap>[125X [27XIdentitySection( C3 );     [127X[104X
    [4X[28XSectionByHomomorphism( s3, g18, [ (4,5,6), (2,3)(5,6) ], [128X[104X
    [4X[28X[ (4,5,6), (2,3)(5,6) ] )[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.1-5 WhiteheadProduct[101X
  
  [33X[1;0Y[29X[2XWhiteheadProduct[102X( [3Xupi[103X, [3Xupj[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XWhiteheadOrder[102X( [3Xup[103X ) [32X operation[133X
  
  [33X[0;0YThe  [10XWhiteheadProduct[110X,  defined  in  section  [14X5.1[114X,  may  be  applied  to two
  derivations  to  form  [22Xχ_i  ⋆ χ_j[122X, or to two sections to form [22Xξ_i ⋆ ξ_j[122X. The
  [10XWhiteheadOrder[110X  of  a regular derivation [22Xχ[122X is the smallest power of [22Xχ[122X, using
  this product, equal to the [2XIdentityDerivation[102X ([14X5.1-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xchi12 := WhiteheadProduct( chi1, chi2 );[127X[104X
    [4X[28XDerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (1,3,2)(4,5,6), () ] )[128X[104X
    [4X[25Xgap>[125X [27Xxi12 := WhiteheadProduct( xi1, xi2 );[127X[104X
    [4X[28XSectionByHomomorphism( s3, g18, [ (4,5,6), (2,3)(5,6) ], [128X[104X
    [4X[28X[ (1,3,2)(4,6,5), (2,3)(5,6) ] )[128X[104X
    [4X[25Xgap>[125X [27Xxi12 = SectionByDerivation( chi12 ); [127X[104X
    [4X[28Xtrue [128X[104X
    [4X[25Xgap>[125X [27X[ WhiteheadOrder( chi2 ), WhiteheadOrder( xi2 ) ];[127X[104X
    [4X[28X[ 2, 2 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X5.2 [33X[0;0YWhitehead Monoids and Groups[133X[101X
  
  [33X[0;0YAs  mentioned  at  the  beginning  of  this  chapter,  the  Whitehead monoid
  [22XDer(calX)[122X  of  [22XcalX[122X is the monoid of all derivations from [22XR[122X to [22XS[122X. Monoids of
  derivations    have    representation   [10XIsMonoidOfUp2DimensionalMappingsObj[110X.
  Multiplication  tables  for  Whitehead  monoids  enable  the construction of
  transformation representations.[133X
  
  [1X5.2-1 AllDerivations[101X
  
  [33X[1;0Y[29X[2XAllDerivations[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XImagesList[102X( [3Xobj[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XDerivationClass[102X( [3Xmon[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XImagesTable[102X( [3Xobj[103X ) [32X attribute[133X
  
  [33X[0;0YUsing  our  example  [10XX3[110X  we  find that there are just nine derivations. Here
  [10XAllDerivations[110X returns a collection of up mappings with attributes:[133X
  
  [30X    [33X[0;6Y[10XObject2d[110X - the crossed modules [22XcalX[122X;[133X
  
  [30X    [33X[0;6Y[10XImagesList[110X  -  a  list,  for  each  derivation,  of  the images of the
        generators of the range group;[133X
  
  [30X    [33X[0;6Y[10XDerivationClass[110X  -  the  string "all"; other classes include "regular"
        and "principal";[133X
  
  [30X    [33X[0;6Y[10XImagesTable[110X  - this is a table whose [22X[i,j][122X-th entry is the position in
        the  list  of  elements of [22XS[122X of the image under the [22Xi[122X-th derivation of
        the [22Xj[122X-th element of [22XR[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xall3 := AllDerivations( X3 );[127X[104X
    [4X[28Xmonoid of derivations with images list:[128X[104X
    [4X[28X[ (), () ][128X[104X
    [4X[28X[ (), (1,3,2)(4,5,6) ][128X[104X
    [4X[28X[ (), (1,2,3)(4,6,5) ][128X[104X
    [4X[28X[ (1,3,2)(4,5,6), () ][128X[104X
    [4X[28X[ (1,3,2)(4,5,6), (1,3,2)(4,5,6) ][128X[104X
    [4X[28X[ (1,3,2)(4,5,6), (1,2,3)(4,6,5) ][128X[104X
    [4X[28X[ (1,2,3)(4,6,5), () ][128X[104X
    [4X[28X[ (1,2,3)(4,6,5), (1,3,2)(4,5,6) ][128X[104X
    [4X[28X[ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ][128X[104X
    [4X[25Xgap>[125X [27XDerivationClass( all3 );[127X[104X
    [4X[28X"all"[128X[104X
    [4X[25Xgap>[125X [27XPerform( ImagesTable( all3 ), Display );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1 ][128X[104X
    [4X[28X[ 1, 1, 1, 3, 3, 3 ][128X[104X
    [4X[28X[ 1, 1, 1, 2, 2, 2 ][128X[104X
    [4X[28X[ 1, 3, 2, 1, 3, 2 ][128X[104X
    [4X[28X[ 1, 3, 2, 3, 2, 1 ][128X[104X
    [4X[28X[ 1, 3, 2, 2, 1, 3 ][128X[104X
    [4X[28X[ 1, 2, 3, 1, 2, 3 ][128X[104X
    [4X[28X[ 1, 2, 3, 3, 1, 2 ][128X[104X
    [4X[28X[ 1, 2, 3, 2, 3, 1 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.2-2 WhiteheadMonoidTable[101X
  
  [33X[1;0Y[29X[2XWhiteheadMonoidTable[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XWhiteheadTransformationMonoid[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0YThe  [10XWhiteheadMonoidTable[110X of [22XcalX[122X is the multiplication table whose [22X[i,j][122X-th
  entry  is the position [22Xk[122X in the list of derivations of the Whitehead product
  [22Xχ_i*χ_j = χ_k[122X.[133X
  
  [33X[0;0YUsing  the  rows  of  the  table  as  transformations,  we may construct the
  [10XWhiteheadTransformationMonoid[110X of [22XcalX[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xwmt3 := WhiteheadMonoidTable( X3 );; [127X[104X
    [4X[25Xgap>[125X [27XPerform( wmt3, Display );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9 ][128X[104X
    [4X[28X[ 2, 3, 1, 5, 6, 4, 8, 9, 7 ][128X[104X
    [4X[28X[ 3, 1, 2, 6, 4, 5, 9, 7, 8 ][128X[104X
    [4X[28X[ 4, 6, 5, 1, 3, 2, 7, 9, 8 ][128X[104X
    [4X[28X[ 5, 4, 6, 2, 1, 3, 8, 7, 9 ][128X[104X
    [4X[28X[ 6, 5, 4, 3, 2, 1, 9, 8, 7 ][128X[104X
    [4X[28X[ 7, 7, 7, 7, 7, 7, 7, 7, 7 ][128X[104X
    [4X[28X[ 8, 8, 8, 8, 8, 8, 8, 8, 8 ][128X[104X
    [4X[28X[ 9, 9, 9, 9, 9, 9, 9, 9, 9 ][128X[104X
    [4X[25Xgap>[125X [27Xwtm3 := WhiteheadTransformationMonoid( X3 );[127X[104X
    [4X[28X<transformation monoid of degree 9 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfMonoid( wtm3 ); [127X[104X
    [4X[28X[ Transformation( [ 2, 3, 1, 5, 6, 4, 8, 9, 7 ] ), [128X[104X
    [4X[28X  Transformation( [ 4, 6, 5, 1, 3, 2, 7, 9, 8 ] ), [128X[104X
    [4X[28X  Transformation( [ 7, 7, 7, 7, 7, 7, 7, 7, 7 ] ) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.2-3 RegularDerivations[101X
  
  [33X[1;0Y[29X[2XRegularDerivations[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XWhiteheadGroupTable[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XWhiteheadPermGroup[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XIsWhiteheadPermGroup[102X( [3XG[103X ) [32X property[133X
  [33X[1;0Y[29X[2XWhiteheadRegularGroup[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XWhiteheadGroupIsomorphism[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0Y[10XRegularDerivations[110X are those derivations which are invertible in the monoid.
  The multiplication table for a Whitehead group - a subtable of the Whitehead
  monoid  table  -  enables  the  construction of a permutation representation
  [10XWhiteheadPermGroup[110X   [22XWcalX[122X  of  [22XcalX[122X.  This  group  satisfies  the  property
  [10XIsWhiteheadPermGroup[110X. and [22XcalX[122X is its attribute [10XObject2d[110X.[133X
  
  [33X[0;0YOf  the nine derivations of [10XX3[110X just six are regular. The associated group is
  isomorphic to the symmetric group [10Xs3[110X.[133X
  
  [33X[0;0YTaking the rows of the [10XWhiteheadGroupTable[110X as permutations, we may construct
  the      [10XWhiteheadRegularGroup[110X      of     [22XcalX[122X.     Then,     seeking     a
  [10XSmallerDegreePermutationRepresentation[110X,         we         obtain        the
  [10XWhiteheadGroupIsomorphism[110X whose image is the [10XWhiteheadPermGroup[110X of [22XcalX[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xreg3 := RegularDerivations( X3 );[127X[104X
    [4X[28Xmonoid of derivations with images list:[128X[104X
    [4X[28X[ (), () ][128X[104X
    [4X[28X[ (), (1,3,2)(4,5,6) ][128X[104X
    [4X[28X[ (), (1,2,3)(4,6,5) ][128X[104X
    [4X[28X[ (1,3,2)(4,5,6), () ][128X[104X
    [4X[28X[ (1,3,2)(4,5,6), (1,3,2)(4,5,6) ][128X[104X
    [4X[28X[ (1,3,2)(4,5,6), (1,2,3)(4,6,5) ][128X[104X
    [4X[25Xgap>[125X [27XDerivationClass( reg3 );[127X[104X
    [4X[28X"regular"[128X[104X
    [4X[25Xgap>[125X [27Xwgt3 := WhiteheadGroupTable( X3 );; [127X[104X
    [4X[25Xgap>[125X [27XPerform( wgt3, Display );[127X[104X
    [4X[28X[ [ 1, 2, 3, 4, 5, 6 ],[128X[104X
    [4X[28X  [ 2, 3, 1, 5, 6, 4 ],[128X[104X
    [4X[28X  [ 3, 1, 2, 6, 4, 5 ],[128X[104X
    [4X[28X  [ 4, 6, 5, 1, 3, 2 ],[128X[104X
    [4X[28X  [ 5, 4, 6, 2, 1, 3 ],[128X[104X
    [4X[28X  [ 6, 5, 4, 3, 2, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xwpg3 := WhiteheadPermGroup( X3 );[127X[104X
    [4X[28XGroup([ (1,2,3), (1,2) ])[128X[104X
    [4X[25Xgap>[125X [27XIsWhiteheadPermGroup( wpg3 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XObject2d( wpg3 );[127X[104X
    [4X[28X[c3->s3][128X[104X
    [4X[25Xgap>[125X [27XWhiteheadRegularGroup( X3 );[127X[104X
    [4X[28XGroup([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ])[128X[104X
    [4X[25Xgap>[125X [27XMappingGeneratorsImages( WhiteheadGroupIsomorphism( X3 ) );[127X[104X
    [4X[28X[ [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ], [ (1,2,3), (1,2) ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.2-4 PrincipalDerivations[101X
  
  [33X[1;0Y[29X[2XPrincipalDerivations[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XPrincipalDerivationSubgroup[102X( [3XX0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XWhiteheadHomomorphism[102X( [3XX0[103X ) [32X attribute[133X
  
  [33X[0;0YThe  principal  derivations  form  a  subgroup  of  the Whitehead group. The
  [10XPrincipalDerivationSubgroup[110X   is   the   corresponding   subgroup   of   the
  [10XWhiteheadPermGroup[110X.[133X
  
  [33X[0;0YThe  Whitehead homomorphism [22Xη : S -> WcalX, s ↦ η_s[122X for [22XcalX[122X maps the source
  group of [22XcalX[122X to the Whitehead group of [22XcalX[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XPDX3 := PrincipalDerivations( X3 );[127X[104X
    [4X[28Xmonoid of derivations with images list:[128X[104X
    [4X[28X[ (), () ][128X[104X
    [4X[28X[ (), (1,3,2)(4,5,6) ][128X[104X
    [4X[28X[ (), (1,2,3)(4,6,5) ][128X[104X
    [4X[25Xgap>[125X [27XPDSX3 := PrincipalDerivationSubgroup( X3 );[127X[104X
    [4X[28XGroup([ (1,2,3) ])[128X[104X
    [4X[25Xgap>[125X [27XWhom3 := WhiteheadHomomorphism( X3 );[127X[104X
    [4X[28X[ (1,2,3)(4,6,5) ] -> [ (1,2,3) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[12XExercise:[112X[22X~[122X  Use  the  two crossed module axioms to show that [22Xη_s_1 ⋆ η_s_2 =
  η_s_1s_2[122X.[133X
  
  
  [1X5.3 [33X[0;0YEndomorphisms determined by a derivation[133X[101X
  
  [1X5.3-1 SourceEndomorphism[101X
  
  [33X[1;0Y[29X[2XSourceEndomorphism[102X( [3Xchi[103X ) [32X operation[133X
  
  [33X[0;0YA  derivation [22Xχ[122X of [22XcalX = (∂ : S -> R)[122X determines an endomorphism [22Xσ_χ : S ->
  S,~ s ↦ s(χ ∂ s)[122X. We may verify that [22Xσ_χ[122X is a homomorphism by:[133X
  
  
  [24X[33X[0;6Y\sigma_{\chi}(s_1s_2)   ~=~  s_1s_2\chi((\partial  s_1)(\partial  s_2))  ~=~
  s_1s_2(\chi\partial     s_1)^{\partial     s_2}(\chi\partial     s_2)    ~=~
  s_1(\chi\partial           s_1)s_2(\chi\partial           s_2)           ~=~
  (\sigma_{\chi}s_1)(\sigma_{\chi}s_2).[133X
  
  [124X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xsigma2 := SourceEndomorphism( chi2 );[127X[104X
    [4X[28X[ (1,2,3)(4,6,5) ] -> [ (1,3,2)(4,5,6) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.3-2 RangeEndomorphism[101X
  
  [33X[1;0Y[29X[2XRangeEndomorphism[102X( [3Xchi[103X ) [32X operation[133X
  
  [33X[0;0YA  derivation [22Xχ[122X of [22XcalX = (∂ : S -> R)[122X determines an endomorphism [22Xρ_χ : R ->
  R,~ r ↦ r(∂ χ r)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xrho2 := RangeEndomorphism( chi2 );[127X[104X
    [4X[28X[ (4,5,6), (2,3)(5,6) ] -> [ (4,6,5), (2,3)(4,6) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X5.3-3 Object2dEndomorphism[101X
  
  [33X[1;0Y[29X[2XObject2dEndomorphism[102X( [3Xchi[103X ) [32X operation[133X
  
  [33X[0;0YA  derivation [22Xχ[122X of [22XcalX = (∂ : S -> R)[122X determines an endomorphism [22Xα_χ : calX
  ->  calX[122X  whose source and range endomorphisms are given by the previous two
  operations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27Xalpha2 := Object2dEndomorphism( chi2 );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( alpha2 );[127X[104X
    [4X[28XMorphism of crossed modules :- [128X[104X
    [4X[28X: Source = [c3->s3] with generating sets:[128X[104X
    [4X[28X  [ (1,2,3)(4,6,5) ][128X[104X
    [4X[28X  [ (4,5,6), (2,3)(5,6) ][128X[104X
    [4X[28X: Range = Source[128X[104X
    [4X[28X: Source Homomorphism maps source generators to:[128X[104X
    [4X[28X  [ (1,3,2)(4,5,6) ][128X[104X
    [4X[28X: Range Homomorphism maps range generators to:[128X[104X
    [4X[28X  [ (4,6,5), (2,3)(4,6) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  
  [1X5.4 [33X[0;0YWhitehead groups for cat[22X^1[122X[101X[1X-groups[133X[101X
  
  [1X5.4-1 AllSections[101X
  
  [33X[1;0Y[29X[2XAllSections[102X( [3XC0[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRegularSections[102X( [3XC0[103X ) [32X attribute[133X
  
  [33X[0;0YThese  operations are currently obtained by running the equivalent operation
  for derivations and then converting the result to sections.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XAllSections( C3 );[127X[104X
    [4X[28Xmonoid of sections with images list:[128X[104X
    [4X[28X[ (4,5,6), (2,3)(5,6) ][128X[104X
    [4X[28X[ (4,5,6), (1,3)(4,5) ][128X[104X
    [4X[28X[ (4,5,6), (1,2)(4,6) ][128X[104X
    [4X[28X[ (1,3,2)(4,6,5), (2,3)(5,6) ][128X[104X
    [4X[28X[ (1,3,2)(4,6,5), (1,3)(4,5) ][128X[104X
    [4X[28X[ (1,3,2)(4,6,5), (1,2)(4,6) ][128X[104X
    [4X[28X[ (1,2,3), (2,3)(5,6) ][128X[104X
    [4X[28X[ (1,2,3), (1,3)(4,5) ][128X[104X
    [4X[28X[ (1,2,3), (1,2)(4,6) ][128X[104X
    [4X[25Xgap>[125X [27XRegularSections( C3 );         [127X[104X
    [4X[28Xmonoid of sections with images list:[128X[104X
    [4X[28X[ (4,5,6), (2,3)(5,6) ][128X[104X
    [4X[28X[ (4,5,6), (1,3)(4,5) ][128X[104X
    [4X[28X[ (4,5,6), (1,2)(4,6) ][128X[104X
    [4X[28X[ (1,3,2)(4,6,5), (2,3)(5,6) ][128X[104X
    [4X[28X[ (1,3,2)(4,6,5), (1,3)(4,5) ][128X[104X
    [4X[28X[ (1,3,2)(4,6,5), (1,2)(4,6) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
