  
  [1X39 [33X[0;0YMiscellaneous[133X[101X
  
  
  [1X39.1 [33X[0;0Y [133X[101X
  
  [1X39.1-1 SL2Z[101X
  
  [33X[1;0Y[29X[2XSL2Z[102X( [3Xp[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSL2Z[102X( [3X1/m[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  prime  [22Xp[122X or the reciprocal [22X1/m[122X of a square free integer [22Xm[122X. In the
  first  case  the  function  returns  the  conjugate [22XSL(2,Z)^P[122X of the special
  linear  group  [22XSL(2,Z)[122X  by the matrix [22XP=[[1,0],[0,p]][122X. In the second case it
  returns the group [22XSL(2,Z[1/m])[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap11.html[107X) ,            2
  ([7X../www/SideLinks/About/aboutArithmetic.html[107X) ,                            3
  ([7X../www/SideLinks/About/aboutIntro.html[107X) [133X
  
  [1X39.1-2 BigStepLCS[101X
  
  [33X[1;0Y[29X[2XBigStepLCS[102X( [3XG[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  group  [22XG[122X  and  a  positive  integer  [22Xn[122X.  It  returns  a subseries
  [22XG=L_1[122X>[22XL_2[122X>[22X... L_k=1[122X of the lower central series of [22XG[122X such that [22XL_i/L_i+1[122X has
  order greater than [22Xn[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X      1      ([7X../www/SideLinks/About/aboutPerformance.html[107X) ,     2
  ([7X../www/SideLinks/About/aboutExtensions.html[107X) [133X
  
  [1X39.1-3 Classify[101X
  
  [33X[1;0Y[29X[2XClassify[102X( [3XL[103X, [3XInv[103X ) [32X function[133X
  
  [33X[0;0YInputs a list of objects [22XL[122X and a function [22XInv[122X which computes an invariant of
  each  object.  It  returns a list of lists which classifies the objects of [22XL[122X
  according to the invariant..[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap5.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,                             3
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) ,                              4
  ([7X../www/SideLinks/About/aboutquasi.html[107X) ,                                 5
  ([7X../www/SideLinks/About/aboutTensorSquare.html[107X) [133X
  
  [1X39.1-4 RefineClassification[101X
  
  [33X[1;0Y[29X[2XRefineClassification[102X( [3XC[103X, [3XInv[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  list  [22XC:=Classify(L,OldInv)[122X  and returns a refined classification
  according to the invariant [22XInv[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X       1      ([7X../www/SideLinks/About/aboutQuandles2.html[107X) ,      2
  ([7X../www/SideLinks/About/aboutQuandles.html[107X) [133X
  
  [1X39.1-5 Compose[101X
  
  [33X[1;0Y[29X[2XCompose[102X( [3Xf[103X, [3Xg[103X ) [32X function[133X
  
  [33X[0;0YInputs   two   [22XFpG[122X-module   homomorphisms   [22Xf:M   ⟶  N[122X  and  [22Xg:L  ⟶  M[122X  with
  [22XSource(f)=Target(g)[122X . It returns the composite homomorphism [22Xfg:L ⟶ N[122X .[133X
  
  [33X[0;0YThis also applies to group homomorphisms [22Xf,g[122X.[133X
  
  [33X[0;0Y[12XExamples:[112X 1 ([7X../www/SideLinks/About/aboutFunctorial.html[107X) [133X
  
  [1X39.1-6 HAPcopyright[101X
  
  [33X[1;0Y[29X[2XHAPcopyright[102X(  ) [32X function[133X
  
  [33X[0;0YThis function provides details of HAP'S GNU public copyright licence.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X39.1-7 IsLieAlgebraHomomorphism[101X
  
  [33X[1;0Y[29X[2XIsLieAlgebraHomomorphism[102X( [3Xf[103X ) [32X function[133X
  
  [33X[0;0YInputs  an  object  [22Xf[122X and returns true if [22Xf[122X is a homomorphism [22Xf:A ⟶ B[122X of Lie
  algebras (preserving the Lie bracket).[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X39.1-8 IsSuperperfect[101X
  
  [33X[1;0Y[29X[2XIsSuperperfect[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  group  [22XG[122X and returns "true" if both the first and second integral
  homology of [22XG[122X is trivial. Otherwise, it returns "false".[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X39.1-9 MakeHAPManual[101X
  
  [33X[1;0Y[29X[2XMakeHAPManual[102X(  ) [32X function[133X
  
  [33X[0;0YThis function creates the manual for HAP from an XML file.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X39.1-10 PermToMatrixGroup[101X
  
  [33X[1;0Y[29X[2XPermToMatrixGroup[102X( [3XG[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0YInputs  a  permutation  group  [22XG[122X  and  its  degree  [22Xn[122X.  Returns  a bijective
  homomorphism [22Xf:G ⟶ M[122X where [22XM[122X is a group of permutation matrices.[133X
  
  [33X[0;0Y[12XExamples:[112X             1             ([7X../tutorial/chap7.html[107X) ,             2
  ([7X../www/SideLinks/About/aboutTwistedCoefficients.html[107X) [133X
  
  [1X39.1-11 SolutionsMatDestructive[101X
  
  [33X[1;0Y[29X[2XSolutionsMatDestructive[102X( [3XM[103X, [3XB[103X ) [32X function[133X
  
  [33X[0;0YInputs  an  [22Xm×n[122X  matrix  [22XM[122X and a [22Xk×n[122X matrix [22XB[122X over a field. It returns a k×m
  matrix [22XS[122X satisfying [22XSM=B[122X.[133X
  
  [33X[0;0YThe  function  will leave matrix [22XM[122X unchanged but will probably change matrix
  [22XB[122X.[133X
  
  [33X[0;0Y(This    is    a    trivial   rewrite   of   the   standard   GAP   function
  [22XSolutionMatDestructive([122X<[22Xmat[122X>,<[22Xvec[122X>) .)[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X39.1-12 LinearHomomorphismsPersistenceMat[101X
  
  [33X[1;0Y[29X[2XLinearHomomorphismsPersistenceMat[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs  a composable sequence [22XL[122X of vector space homomorphisms. It returns an
  integer  matrix  containing  the  dimensions  of  the  images of the various
  composites. The sequence [22XL[122X is determined up to isomorphism by this matrix.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X39.1-13 NormalSeriesToQuotientHomomorphisms[101X
  
  [33X[1;0Y[29X[2XNormalSeriesToQuotientHomomorphisms[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YInputs  an  (increasing  or  decreasing) chain [22XL[122X of normal subgroups in some
  group  [22XG[122X.  This [22XG[122X is the largest group in the chain. It returns the sequence
  of composable group homomorphisms [22XG/L[i] → G/L[i+/-1][122X.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
  [1X39.1-14 TestHap[101X
  
  [33X[1;0Y[29X[2XTestHap[102X(  ) [32X function[133X
  
  [33X[0;0YThis  runs  a  representative sample of HAP functions and checks to see that
  they produce the correct output.[133X
  
  [33X[0;0Y[12XExamples:[112X[133X
  
