Use GroupHomset_libgap for permutation groups, as well

[soehms]

At the moment the construction of group homomorphisms between groups implemented under the classes ParentLibGAP and PermutationGroup_generic (and vice versa) is not possible in sage using the method hom. For example, it isn't possible to construct the natural projection from symplectic groups onto the corresponding projective group although this is possible using GAP:

sage: Sp43 = Sp(4,3)
sage: PSp43 = PSp(4,3)
sage: natProj = Sp43.hom(PSp43.gens())
Traceback (most recent call last):
...
TypeError: unable to convert [(3,4)(6,7)(9,10)(12,13)(17,20)(18,21)(19,22)(23,32)(24,33)(25,34)(26,38)(27,39)(28,40)(29,35)(30,36)(31,37), (1,5,14,17,27,22,19,36,3)(2,6,32)(4,7,23,20,37,13,16,26,40)(8,24,29,30,39,10,33,11,34)(9,15,35)(12,25,38)(21,28,31)] to an element of Set of Morphisms from Symplectic Group of degree 4 over Finite Field of size 3 to The projective symplectic linear group of degree 4 over Finite Field of size 3 in Category of finite groups

The reason for this is that the constructor of the class GroupHomset_libgap explicitly checks both groups to be instances of ParentLibGAP. So an easy way to have hom work in such cases, as well, would be to allow PermutationGroup_generic additionally.

An alternative option is to shift PermutationGroup_generic into the ParentLibGAP framework. But this would cause a lot of work to have all doctests pass through. The main problem here is that the element class of PermutationGroup_generic still has parent as second (optional) argument (in opposite to ParentLibGAP).

Therefore, this ticket will follow the first option!

Depends on #26420

CC: @tscrim @simonbrandhorst

Component: group theory

Keywords: permutation group, homomorphism, libgap

Author: Sebastian Oehms

Branch: 66ceb94

Reviewer: Simon Brandhorst

Issue created by migration from https://trac.sagemath.org/ticket/26750

[soehms]

Branch: u/soehms/libgap_morphism_permgroup-26750

[soehms]

Dependencies: #26420

[soehms]

Commit: 28f39df

[soehms]

comment:2

I do the following:

1. I add the method gap to the classes PermutationGroup_generic and PermutationGroupElement to achieve compatibility with class GroupHomset_libgap and its elements class. For the same reason I add the method _subgroup_constructor to PermutationGroup_generic (needed in the methods pushforward, kernel and preimage in GroupMorphism_libgap)

2. I implement the method _Hom_ in PermutationGroup_generic to have hom use class GroupHomset_libgap.

3. In the module libgap_morphism I add PermutationGroup_generic to the list of allowed parent-instances at all relevant pieces of code. Furthermore, I changed the _element_constructor_ in order to avoid buffer size overflow by transferring long lists of large permutation generators to GAP. This is done in a new local method _obtain_gap_hom.

4. I take the occasion to implement two methods from framework: section of class Map and natural_map of class HomsetWithBase improving their default implementation using the possibilities of GroupHomset_libgap and its element class.

---

New commits:

841a405	group morphism constructor from a libgap morphism
08a0735	a doctest
b2d9622	docs
21cb664	correct docstring for argument `gap_hom`
2a432d0	Merge commit '21cb66449cd424f94d15bd695fa986a0ce40c30c' of git://trac.sagemath.org/sage into libgap_morphism_permgroup-26750
28f39df	initial version

[simonbrandhorst]

comment:3

I am not convinced the function _obtain_gap_hom is useful/necessary.
It seems to be used only for the natural_map method. But then the code just belongs there.
But indeed one could replace

gens = [g.gap() for g in dom.gens()]

by

gens = dom.gap().GeneratorsOfGroup()

in the _element_constructor_
and likewise for the images in natural map. Would that fix the overflow?

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

deaa647

remove unnecessary method _obtain_gap_hom

[sagetrac-git]

Changed commit from 28f39df to deaa647

[soehms]

comment:5

Thanks for looking at this, Simon! Of course you are right: This method has been a stupid idea! I corrected as you suggested.

[sagetrac-git]

Branch pushed to git repo; I updated commit sha1. New commits:

0133335	implement subgroup / ambient methods for MatrixGroup_base
747bc16	removed unused keyword
c3a92a4	Merge branch 'develop' into matrix_groups_subgroups-25894
901441b	Merge branch 'develop' into matrix_groups_subgroups-25894
b02eb31	Merge branch 'libgap_morphism_permgroup-26750' into cubic_braid_group
be4674b	initial implementation
6a07854	Merge branch 'develop' into adapt_unitary_burau-26657
806d0c6	Merge branch 'adapt_unitary_burau-26657' into cubic_braid_group
464b784	Merge branch 'libgap_morphism_permgroup-26750' into cubic_braid_group
2b74ce4	restore deleted doctest

[sagetrac-git]

Changed commit from deaa647 to 2b74ce4