# GAP Integration

This section explains how Oscar interacts with GAP.

## The Julia package GAP.jl

This package provides a bidirectional interface between GAP and Julia. Its documentation describes how to call GAP functions in Julia code and vice versa, and how low level Julia objects can be converted to GAP objects and vice versa.

When one works interactively in an Oscar session, calling `GAP.prompt()`

opens a GAP session which has access to the variables in the Julia session, in particular to all Oscar functions and objects; one can return to the Julia prompt by entering `quit;`

in the GAP session.

## Interface functionalities beyond GAP.jl

For code involving Julia types that are defined in Oscar, GAP.jl cannot provide utility functions such as conversions to and from GAP.

The GAP package OscarInterface (at

`gap/OscarInterface`

) is intended to contain the GAP code in question, for example the declarations of new filters and the installation of new methods.Note that such code must be loaded at runtime into the GAP session that is started by Julia, and the OscarInterface package gets loaded in Oscar's

`__init__`

function.The files in the directory

`src/GAP`

are intended to contain the Julia code in question, for example conversions from GAP to`ZZRingElem`

,`QQFieldElem`

,`FinFieldElem`

, etc., and the construction of isomorphisms between algebraic structures such as rings and fields in GAP and Oscar, via`Oscar.iso_oscar_gap`

and`Oscar.iso_gap_oscar`

.In Oscar code, global GAP variables can be accessed as members of

`GAP.Globals`

, but for the case of GAP functions, it is more efficient to use`Oscar.GAPWrap`

instead.For example, if one wants to call GAP's

`IsFinite`

then it is recommended to replace the call`GAP.Globals.IsFinite(x)::Bool`

, for some GAP object`x`

(a group or a ring or a list, etc.), by`Oscar.GAPWrap.IsFinite(x)`

. This works only if the method in question gets defined in`src/GAP/wrappers.jl`

, thus methods with the required signatures should be added to this file when they turn out to be needed.(The reason why we collect the

`GAP.@wrap`

lines in an Oscar file and not inside GAP.jl is that we can extend the list without waiting for releases of GAP.jl.)In GAP code, global Julia variables can be accessed as members of

`Julia`

, relative to its`Main`

module. For example, one can call`Julia.sqrt`

and`Julia.typeof`

(or`Julia.Base.sqrt`

and`Julia.Core.typeof`

) in GAP code.In order to access variables from the

`Oscar`

module, it is not safe to use`Julia.Oscar`

because the module`Oscar`

is not always defined in`Main`

. Instead, there is the global GAP variable`Oscar`

.

`iso_oscar_gap`

— Function`Oscar.iso_oscar_gap(R) -> Map{T, GapObj}`

Return an isomorphism `f`

with domain `R`

and `codomain`

a GAP object `S`

.

Elements `x`

of `R`

are mapped to `S`

via `f(x)`

, and elements `y`

of `S`

are mapped to `R`

via `preimage(f, y)`

.

Matrices `m`

over `R`

are mapped to matrices over `S`

via `map_entries(f, m)`

, and matrices `n`

over `S`

are mapped to matrices over `R`

via `Oscar.preimage_matrix(f, n)`

.

Admissible values of `R`

and the corresponding `S`

are currently as follows.

`R` | `S` (in `GAP.Globals` ) |
---|---|

`ZZ` | `Integers` |

`QQ` | `Rationals` |

`residue_ring(ZZ, n)[1]` | `mod(Integers, n)` |

`finite_field(p, d)[1]` | `GF(p, d)` |

`cyclotomic_field(n)[1]` | `CF(n)` |

`number_field(f::QQPolyRingElem)[1]` | `AlgebraicExtension(Rationals, g)` |

`abelian_closure(QQ)[1]` | `Cyclotomics` |

`polynomial_ring(F)[1]` | `PolynomialRing(G)` |

`polynomial_ring(F, n)[1]` | `PolynomialRing(G, n)` |

(Here `g`

is the polynomial over `GAP.Globals.Rationals`

that corresponds to `f`

, and `G`

is equal to `Oscar.iso_oscar_gap(F)`

.)

**Examples**

```
julia> f = Oscar.iso_oscar_gap(ZZ);
julia> x = ZZ(2)^100; y = f(x)
GAP: 1267650600228229401496703205376
julia> preimage(f, y) == x
true
julia> m = matrix(ZZ, 2, 3, [1, 2, 3, 4, 5, 6]);
julia> n = map_entries(f, m)
GAP: [ [ 1, 2, 3 ], [ 4, 5, 6 ] ]
julia> Oscar.preimage_matrix(f, n) == m
true
julia> R, x = polynomial_ring(QQ);
julia> f = Oscar.iso_oscar_gap(R);
julia> pol = x^2 + x - 1;
julia> y = f(pol)
GAP: x_1^2+x_1-1
julia> preimage(f, y) == pol
true
```

The functions `Oscar.iso_oscar_gap`

and `Oscar.iso_gap_oscar`

are not injective. Due to caching, it may happen that `S`

stores an attribute value of `Oscar.iso_gap_oscar(S)`

, but that the codomain of this map is not identical with or even not equal to the given `R`

.

Note also that `R`

and `S`

may differ w.r.t. some structural properties because GAP does not support all kinds of constructions that are possible in Oscar. For example, if `R`

is a non-simple number field then `S`

will be a simple extension because GAP knows only simple field extensions. Thus using `Oscar.iso_oscar_gap(R)`

for objects `R`

whose recursive structure is not fully supported in GAP will likely cause overhead at runtime.

`iso_gap_oscar`

— Function`Oscar.iso_gap_oscar(R) -> Map{GapObj, T}`

Return an isomorphism `f`

with `domain`

the GAP object `R`

and `codomain`

an Oscar object `S`

.

Elements `x`

of `R`

are mapped to `S`

via `f(x)`

, and elements `y`

of `S`

are mapped to `R`

via `preimage(f, y)`

.

Matrices `m`

over `R`

are mapped to matrices over `S`

via `map_entries(f, m)`

, and matrices `n`

over `S`

are mapped to matrices over `R`

via `Oscar.preimage_matrix(f, n)`

.

Admissible values of `R`

and the corresponding `S`

are currently as follows.

`S` (in `GAP.Globals` ) | `R` |
---|---|

`Integers` | `ZZ` |

`Rationals` | `QQ` |

`mod(Integers, n)` | `residue_ring(ZZ, n)[1]` |

`GF(p, d)` | `finite_field(p, d)[1]` |

`CF(n)` | `cyclotomic_field(n)[1]` |

`AlgebraicExtension(Rationals, f)` | `number_field(g)[1]` |

`Cyclotomics` | `abelian_closure(QQ)[1]` |

`PolynomialRing(F)` | `polynomial_ring(G)[1]` |

`PolynomialRing(F, n)` | `polynomial_ring(G, n)[1]` |

(Here `g`

is the polynomial over `QQ`

that corresponds to the polynomial `f`

, and `G`

is equal to `Oscar.iso_gap_oscar(F)`

.)

**Examples**

```
julia> f = Oscar.iso_gap_oscar(GAP.Globals.Integers);
julia> x = ZZ(2)^100; y = preimage(f, x)
GAP: 1267650600228229401496703205376
julia> f(y) == x
true
julia> m = matrix(ZZ, 2, 3, [1, 2, 3, 4, 5, 6]);
julia> n = Oscar.preimage_matrix(f, m)
GAP: [ [ 1, 2, 3 ], [ 4, 5, 6 ] ]
julia> map_entries(f, n) == m
true
julia> R = GAP.Globals.PolynomialRing(GAP.Globals.Rationals);
julia> f = Oscar.iso_gap_oscar(R);
julia> x = gen(codomain(f));
julia> pol = x^2 + x + 1;
julia> y = preimage(f, pol)
GAP: x_1^2+x_1+1
julia> f(y) == pol
true
```

The functions `Oscar.iso_gap_oscar`

and `Oscar.iso_oscar_gap`

are not injective. Due to caching, it may happen that `S`

stores an attribute value of `Oscar.iso_oscar_gap(S)`

, but that the codomain of this map is not identical with or even not equal to the given `R`

.