GAP (XMod) - Chapter 12: Interaction with HAP

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12 Interaction with HAP
12.1 Calling HAP functions
12.1-1 SmallCat1Group
12.1-2 CatOneGroupToXMod
12.1-3 IdCat1Group

### 12 Interaction with HAP

This chapter describes functions which allow functions in the package HAP to be called from XMod.

#### 12.1 Calling HAP functions

In HAP a cat\(^1\)-group is called a `CatOneGroup` and the traditional terms source and target are used for the `TailMap` and `HeadMap`. A `CatOneGroup` is a record `C` with fields `C!.sourceMap` and `C!.targetMap`.

##### 12.1-1 SmallCat1Group

`‣ SmallCat1Group`( n, i, j ) | ( operation )

This operation calls the HAP function `SmallCatOneGroup(n,i,j)` which returns a `CatOneGroup` from the HAP database. This is then converted into an XMod cat\(^1\)-group. Note that the numbering is not the same as that used by the XMod operation `Cat1Select`. In the example `C12` is the converted form of `H12`.

    gap> H12 := SmallCatOneGroup( 12, 4, 3 );
    Cat-1-group with underlying group Group( [ f1, f2, f3 ] ) .
    gap> C12 := SmallCat1Group( 12, 4, 3 );
    [Group( [ f1, f2, f3 ] )=>Group( [ f1, f2, <identity> of ... ] )]

##### 12.1-2 CatOneGroupToXMod

`‣ CatOneGroupToXMod`( C ) | ( operation )

`‣ Cat1GroupToHAP`( C ) | ( operation )

These two functions convert between the two alternative implementations.

    gap> C12 := CatOneGroupToXMod( H12 );
    [Group( [ f1, f2, f3 ] )=>Group( [ f1, f2, <identity> of ... ] )]
    gap> C18 := Cat1Select( 18, 4, 3 );
    [(C3 x C3) : C2=>Group( [ f1, <identity> of ..., f3 ] )]
    gap> H18 := Cat1GroupToHAP( C18 );
    Cat-1-group with underlying group (C3 x C3) : C2 .

##### 12.1-3 IdCat1Group

`‣ IdCat1Group`( C ) | ( operation )

This function calls the HAP function `IdCatOneGroup` on a cat\(^1\)-group \(C\). This returns \([n,i,j]\) if the cat\(^1\)-group is the \(j\)-th structure on the `SmallGroup(n,i)`.

    gap> IdCatOneGroup( H18 );
    [ 18, 4, 4 ]
    gap> IdCat1Group( C18 );
    [ 18, 4, 4 ]

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