  
  [1X37 [33X[0;0YMiscellaneous[133X[101X
  
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XSL2Z(p) [110X[133X
  [33X[0;0Y[10XSL2Z(1/m) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XBigStepLCS(G,n) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XClassify(L,Inv) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XRefineClassification(C,Inv) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XCompose(f,g) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XHAPcopyright() [110X[133X
  
  [33X[0;0YThis function provides details of HAP'S GNU public copyright licence.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XIsLieAlgebraHomomorphism(f) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XIsSuperperfect(G) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XMakeHAPManual()[110X[133X
  
  [33X[0;0YThis function creates the manual for HAP from an XML file.[133X
  [33X[0;0Y::::::::::::::::::::::::[133X
  [33X[0;0Y[10XPermToMatrixGroup(G,n) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XSolutionsMatDestructive(M,B) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XLinearHomomorphismsPersistenceMat(L) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XNormalSeriesToQuotientHomomorphisms(L) [110X[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::::::::::::::::::::::::[133X
  [33X[0;0Y[10XTestHap() [110X[133X
  
  [33X[0;0YThis  runs  a  representative sample of HAP functions and checks to see that
  they produce the correct output.[133X
  
