GAP (HAP) - Chapter 34: Miscellaneous

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34 Miscellaneous

### 34 Miscellaneous

`SL2Z(p)` `SL2Z(1/m)` Inputs a prime p or the reciprocal 1/m of a square free integer m. In the first case the function returns the conjugate SL(2,Z)^P of the special linear group SL(2,Z) by the matrix P=[[1,0],[0,p]]. In the second case it returns the group SL(2,Z[1/m]).
`BigStepLCS(G,n)` Inputs a group G and a positive integer n. It returns a subseries G=L_1>L_2>... L_k=1 of the lower central series of G such that L_i/L_i+1 has order greater than n.
`Classify(L,Inv)` Inputs a list of objects L and a function Inv which computes an invariant of each object. It returns a list of lists which classifies the objects of L according to the invariant..
`RefineClassification(C,Inv)` Inputs a list C:=Classify(L,OldInv) and returns a refined classification according to the invariant Inv.
`Compose(f,g)` Inputs two FpG-module homomorphisms f:M ⟶ N and g:L ⟶ M with Source(f)=Target(g) . It returns the composite homomorphism fg:L ⟶ N . This also applies to group homomorphisms f,g.
`HAPcopyright()` This function provides details of HAP'S GNU public copyright licence.
`IsLieAlgebraHomomorphism(f)` Inputs an object f and returns true if f is a homomorphism f:A ⟶ B of Lie algebras (preserving the Lie bracket).
`IsSuperperfect(G)` Inputs a group G and returns "true" if both the first and second integral homology of G is trivial. Otherwise, it returns "false".
`MakeHAPManual()` This function creates the manual for HAP from an XML file.
`PermToMatrixGroup(G,n)` Inputs a permutation group G and its degree n. Returns a bijective homomorphism f:G ⟶ M where M is a group of permutation matrices.
`SolutionsMatDestructive(M,B)` Inputs an m×n matrix M and a k×n matrix B over a field. It returns a k×m matrix S satisfying SM=B. The function will leave matrix M unchanged but will probably change matrix B. (This is a trivial rewrite of the standard GAP function SolutionMatDestructive(<mat>,<vec>) .)
`LinearHomomorphismsPersistenceMat(L)` Inputs a composable sequence L of vector space homomorphisms. It returns an integer matrix containing the dimensions of the images of the various composites. The sequence L is determined up to isomorphism by this matrix.
`NormalSeriesToQuotientHomomorphisms(L)` Inputs an (increasing or decreasing) chain L of normal subgroups in some group G. This G is the largest group in the chain. It returns the sequence of composable group homomorphisms G/L[i] → G/L[i+/-1].
`TestHap()` This runs a representative sample of HAP functions and checks to see that they produce the correct output.

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