<Chapter><Heading> Miscellaneous</Heading> <Section><Heading> &nbsp;</Heading> 
<ManSection> <Func Name="SL2Z" Arg="p"/> <Func Name="SL2Z" Arg="1/m"/> <Description> <P/> Inputs a prime <M>p</M> or the reciprocal <M>1/m</M> of a square free integer <M>m</M>. In the first case the function returns the conjugate <M>SL(2,Z)^P</M> of the special linear group <M>SL(2,Z)</M> by the matrix <M>P=[[1,0],[0,p]]</M>. In the second case it returns the group <M>SL(2,Z[1/m])</M>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap11.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap13.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutArithmetic.html</Link><LinkText>3</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutIntro.html</Link><LinkText>4</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="BigStepLCS" Arg="G,n"/> <Description> <P/> Inputs a group <M>G</M> and a positive integer <M>n</M>. It returns a subseries <M>G=L_1</M>&tgt;<M>L_2</M>&tgt;<M> \ldots L_k=1</M> of the lower central series of <M>G</M> such that <M>L_i/L_{i+1}</M> has order greater than <M>n</M>. <P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutPerformance.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutExtensions.html</Link><LinkText>2</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="Classify" Arg="L,Inv"/> <Description> <P/> Inputs a list of objects <M>L</M> and a function <M>Inv</M> which computes an invariant of each object. It returns a list of lists which classifies the objects of <M>L</M> according to the invariant.. <P/><B>Examples:</B> <URL><Link>../tutorial/chap1.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap4.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap5.html</Link><LinkText>3</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap12.html</Link><LinkText>4</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>5</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>6</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutquasi.html</Link><LinkText>7</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutTensorSquare.html</Link><LinkText>8</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="RefineClassification" Arg="C,Inv"/> <Description> <P/> Inputs a list <M>C:=Classify(L,OldInv)</M> and returns a refined classification according to the invariant <M>Inv</M>. <P/><B>Examples:</B> <URL><Link>../tutorial/chap4.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../tutorial/chap12.html</Link><LinkText>2</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutQuandles2.html</Link><LinkText>3</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutQuandles.html</Link><LinkText>4</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="Compose" Arg="f,g"/> <Description> <P/> Inputs two <M>FpG</M>-module homomorphisms <M> f:M \longrightarrow N</M> and <M>g:L \longrightarrow M</M> with <M>Source(f)=Target(g)</M> . It returns the composite homomorphism <M>fg:L \longrightarrow N</M> . <P/> This also applies to group homomorphisms <M>f,g</M>. <P/><B>Examples:</B> <URL><Link>../www/SideLinks/About/aboutFunctorial.html</Link><LinkText>1</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="HAPcopyright" Arg=""/> <Description> <P/> This function provides details of HAP'S GNU public copyright licence. <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="IsLieAlgebraHomomorphism" Arg="f"/> <Description> <P/> Inputs an object <M>f</M> and returns true if <M>f</M> is a homomorphism <M>f:A \longrightarrow B</M> of Lie algebras (preserving the Lie bracket). <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="IsSuperperfect" Arg="G"/> <Description> <P/> Inputs a group <M>G</M> and returns "true" if both the first and second integral homology of <M>G</M> is trivial. Otherwise, it returns "false". <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="MakeHAPManual" Arg=""/> <Description> <P/> This function creates the manual for HAP from an XML file. <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="PermToMatrixGroup" Arg="G,n"/> <Description> <P/> Inputs a permutation group <M>G</M> and its degree <M>n</M>. Returns a bijective homomorphism <M>f:G \longrightarrow M</M> where <M>M</M> is a group of permutation matrices. <P/><B>Examples:</B> <URL><Link>../tutorial/chap7.html</Link><LinkText>1</LinkText></URL>&nbsp;, <URL><Link>../www/SideLinks/About/aboutTwistedCoefficients.html</Link><LinkText>2</LinkText></URL>&nbsp;
</Description> </ManSection> 
<ManSection> <Func Name="SolutionsMatDestructive" Arg="M,B"/> <Description> <P/> Inputs an <M>m×n</M> matrix <M>M</M> and a <M>k×n</M> matrix <M>B</M> over a field. It returns a k×m matrix <M>S</M> satisfying <M>SM=B</M>. <P/> The function will leave matrix <M>M</M> unchanged but will probably change matrix <M>B</M>. <P/> (This is a trivial rewrite of the standard GAP function <M>SolutionMatDestructive(</M>&tlt;<M>mat</M>&tgt;,&tlt;<M>vec</M>&tgt;) .) <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="LinearHomomorphismsPersistenceMat" Arg="L"/> <Description> <P/> Inputs a composable sequence <M>L</M> of vector space homomorphisms. It returns an integer matrix containing the dimensions of the images of the various composites. The sequence <M>L</M> is determined up to isomorphism by this matrix. <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="NormalSeriesToQuotientHomomorphisms" Arg="L"/> <Description> <P/> Inputs an (increasing or decreasing) chain <M>L</M> of normal subgroups in some group <M>G</M>. This <M>G</M> is the largest group in the chain. It returns the sequence of composable group homomorphisms <M>G/L[i] \rightarrow G/L[i+/-1]</M>. <P/><B>Examples:</B> 
</Description> </ManSection> 
<ManSection> <Func Name="TestHap" Arg=""/> <Description> <P/> This runs a representative sample of HAP functions and checks to see that they produce the correct output. <P/><B>Examples:</B> 
</Description> </ManSection> </Section> </Chapter>