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GAP 4.8.9 installation with standard packages -- copy to your CoCalc project to get it

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<Chapter><Heading> Miscellaneous</Heading>

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<Table Align="|l|" >

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<Row>

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<Item>

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<Index> SL2Z </Index>

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<C>SL2Z(p) </C>

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<C>SL2Z(1/m) </C>

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<P/>

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Inputs a prime <M>p</M> or the reciprocal <M>1/m</M> of  a 

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 square free integer <M>m</M>. In the first case the function

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 returns the conjugate <M>SL(2,Z)^P</M> of the special linear group <M>SL(2,Z)</M>

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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>. 

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> BigStepLCS </Index>

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<C>BigStepLCS(G,n) </C>

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<P/>

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Inputs a group <M>G</M> and a positive integer <M>n</M>. 

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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 

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<M>G</M> such that <M>L_i/L_{i+1}</M> has order greater than <M>n</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> Classify </Index>

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<C>Classify(L,Inv) </C>

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<P/>

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Inputs a list of objects <M>L</M> and a function <M>Inv</M> which

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computes an invariant of each object. It returns a list of lists which classifies the objects of <M>L</M> according to the invariant..

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> RefineClassification </Index>

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<C>RefineClassification(C,Inv) </C>

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<P/>

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Inputs a list <M>C:=Classify(L,OldInv)</M> and returns a refined classification according to the invariant <M>Inv</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> Compose(f,g)</Index>

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<C>Compose(f,g) </C>

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<P/>

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Inputs two <M>FpG</M>-module homomorphisms <M>

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f:M \longrightarrow N</M> and <M>g:L \longrightarrow M</M>

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with <M>Source(f)=Target(g)</M> . 

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It returns the composite homomorphism <M>fg:L \longrightarrow N</M> .

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<P/>

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This also applies to group homomorphisms <M>f,g</M>.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> HAPcopyright</Index>

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<C>HAPcopyright() </C>

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<P/>

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This function provides details of HAP'S GNU public copyright licence.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> IsLieAlgebraHomomorphism</Index>

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<C>IsLieAlgebraHomomorphism(f) </C>

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<P/>

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Inputs an object <M>f</M> and returns true if <M>f</M>

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is a homomorphism <M>f:A \longrightarrow B</M>

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of Lie algebras (preserving the Lie bracket).

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> IsSuperperfect</Index>

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<C>IsSuperperfect(G) </C>

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<P/>

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Inputs a group <M>G</M> and returns "true" if  both the 

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first and second integral homology of <M>G</M> is trivial. 

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Otherwise, it returns "false". 

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</Item>

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</Row>

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<Row>

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<Item>

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<Index>MakeHAPManual</Index>

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<C>MakeHAPManual()</C>

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<P/>

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This function creates the manual for HAP from an XML file.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> PermToMatrixGroup </Index>

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<C>PermToMatrixGroup(G,n) </C>

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<P/>

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Inputs a permutation group <M>G</M> and its degree <M>n</M>. 

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Returns a bijective homomorphism <M>f:G \longrightarrow M</M> where 

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<M>M</M> is a group of permutation matrices.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> SolutionsMatDestructive</Index>

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<C>SolutionsMatDestructive(M,B) </C>

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<P/>

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Inputs an <M>m�n</M> matrix <M>M</M> and a <M>k�n</M> matrix 

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<M>B</M> over a field. It returns a k�m matrix <M>S</M> satisfying 

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<M>SM=B</M>.

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<P/>

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The function will leave matrix <M>M</M> unchanged but 

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will probably change matrix <M>B</M>.

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<P/>

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(This is  a trivial rewrite of the standard GAP function 

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<M>SolutionMatDestructive(</M>&tlt;<M>mat</M>&tgt;,&tlt;<M>vec</M>&tgt;) .) 

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> LinearHomomorphismsPersistenceMat</Index>

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<C>LinearHomomorphismsPersistenceMat(L) </C>

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<P/>

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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.

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> NormalSeriesToQuotientHomomorphisms</Index>

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<C>NormalSeriesToQuotientHomomorphisms(L) </C>

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<P/>

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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

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 returns the sequence of composable group homomorphisms <M>G/L[i] \rightarrow G/L[i+/-1]</M>. 

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</Item>

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</Row>

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<Row>

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<Item>

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<Index> TestHap</Index>

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<C>TestHap() </C>

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<P/>

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This runs a representative sample of HAP functions and checks to see that they produce the correct output.  

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</Item>

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</Row>

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</Table>

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</Chapter>

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