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

##### 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>... 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

##### 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

##### 39.1-5 Compose

`‣ Compose`( f, g ) | ( function )

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.

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 ⟶ 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 ⟶ 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] → 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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