GAP (HAP commands) - Chapter 39: Miscellaneous

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39 Miscellaneous
39.1
39.1-1 SL2Z
39.1-2 BigStepLCS
39.1-3 Classify
39.1-4 RefineClassification
39.1-5 Compose
39.1-6 HAPcopyright
39.1-7 IsLieAlgebraHomomorphism
39.1-8 IsSuperperfect
39.1-9 MakeHAPManual
39.1-10 PermToMatrixGroup
39.1-11 SolutionsMatDestructive
39.1-12 LinearHomomorphismsPersistenceMat
39.1-13 NormalSeriesToQuotientHomomorphisms
39.1-14 TestHap

### 39 Miscellaneous

#### 39.1

##### 39.1-1 SL2Z

`‣ SL2Z`( p ) | ( function )

`‣ SL2Z`( 1/m ) | ( function )

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 \(SL(2,Z[1/m])\).

Examples: 1 , 2 , 3 , 4

##### 39.1-2 BigStepLCS

`‣ BigStepLCS`( G, n ) | ( function )

Inputs a group \(G\) and a positive integer \(n\). It returns a subseries \(G=L_1\)>\(L_2\)>\( \ldots L_k=1\) of the lower central series of \(G\) such that \(L_i/L_{i+1}\) has order greater than \(n\).

Examples: 1 , 2

##### 39.1-3 Classify

`‣ Classify`( L, Inv ) | ( function )

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..

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10

##### 39.1-4 RefineClassification

`‣ RefineClassification`( C, Inv ) | ( function )

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

Examples: 1 , 2 , 3 , 4

##### 39.1-5 Compose

`‣ Compose`( f, g ) | ( function )

Inputs two \(FpG\)-module homomorphisms \( f:M \longrightarrow N\) and \(g:L \longrightarrow M\) with \(Source(f)=Target(g)\) . It returns the composite homomorphism \(fg:L \longrightarrow N\) .

This also applies to group homomorphisms \(f,g\).

Examples: 1

##### 39.1-6 HAPcopyright

`‣ HAPcopyright`( ) | ( function )

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

Examples:

##### 39.1-7 IsLieAlgebraHomomorphism

`‣ IsLieAlgebraHomomorphism`( f ) | ( function )

Inputs an object \(f\) and returns true if \(f\) is a homomorphism \(f:A \longrightarrow B\) of Lie algebras (preserving the Lie bracket).

Examples:

##### 39.1-8 IsSuperperfect

`‣ IsSuperperfect`( G ) | ( function )

Inputs a group \(G\) and returns "true" if both the first and second integral homology of \(G\) is trivial. Otherwise, it returns "false".

Examples:

##### 39.1-9 MakeHAPManual

`‣ MakeHAPManual`( ) | ( function )

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

Examples:

##### 39.1-10 PermToMatrixGroup

`‣ PermToMatrixGroup`( G, n ) | ( function )

Inputs a permutation group \(G\) and its degree \(n\). Returns a bijective homomorphism \(f:G \longrightarrow M\) where \(M\) is a group of permutation matrices.

Examples: 1 , 2

##### 39.1-11 SolutionsMatDestructive

`‣ SolutionsMatDestructive`( M, B ) | ( function )

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 \(SolutionMatDestructive(\)<\(mat\)>,<\(vec\)>) .)

Examples:

##### 39.1-12 LinearHomomorphismsPersistenceMat

`‣ LinearHomomorphismsPersistenceMat`( L ) | ( function )

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.

Examples:

##### 39.1-13 NormalSeriesToQuotientHomomorphisms

`‣ NormalSeriesToQuotientHomomorphisms`( L ) | ( function )

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 \(G/L[i] \rightarrow G/L[i+/-1]\).

Examples:

##### 39.1-14 TestHap

`‣ TestHap`( ) | ( function )

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

Examples:

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