Allow inducing G-sets along group homomorphisms

[lgoettgens]

As discussed earlier today with @ThomasBreuer. I tried to align the name and argument order with the methods for GAPClassFunction and GModule.
Due to not all group types having usable morphism types, I added an undocumented method for general maps between groups, which should get adapted once we have proper group homes.

@jamesnohilly this allows for a similar cleanup of action_homomorphism in #4609 as forFinGenAbGroups in this PR. Depending on the order these two PRs get merged, the later one should perform the cleanup.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 84.92%. Comparing base (fffc6ad) to head (9c75ddf).
Report is 1 commits behind head on master.

Additional details and impacted files

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- Coverage   84.93%   84.92%   -0.01%     
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Files with missing lines	Coverage Δ
src/Groups/GrpAb.jl	93.28% <100.00%> (-0.11%)	⬇️
src/Groups/action.jl	98.00% <100.00%> (+0.09%)	⬆️
src/Groups/gsets.jl	94.27% <100.00%> (+0.22%)	⬆️

... and 1 file with indirect coverage changes

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[ThomasBreuer]

In the example

G = symmetric_group(4)
Omega = gset(G, permuted, [[1,1,2,3]])
H = permutation_group(8, [cperm([1,3], [2,4]), cperm([1,5], [2,6], [3,7], [4,8])])
phi = hom(H, G, [cperm([1,2]), cperm([1,3], [2,4])])
Omega2 = induce(Omega, phi)

my interpretation of the documentation is that Omega and Omega2 shall be equal as sets, they just carry different actions. G acts on Omega, H acts on Omega2.
For that, we cannot take just the seeds of Omega (the points whose closure under the action of G is Omega) as the seeds of Omega2, because H can be (and in the example is) smaller.
(If phi is known to be surjective then taking the seeds of Omega as the seeds of Omega2 is sufficient.
The check whether acting_group(Omega) equals codomain(phi) does not guarantee surjectivity.)

We could "unpack" Omega by computing all its elements, and take them as the seeds for the intended H-set.
If that is not desirable, we could think of creating the orbits of phi(H) on Omega, and then taking orbit representatives as seeds for Omega2. (Then we can also set the flag that Omega2 is closed under the action.)

(If we are instead interested in an H-orbit of a particular element omega of Omega, w.r.t. the action induced by phi, then this is something different. For that, we can first create the action function for H from phi and the action function of Omega, and then create the H-orbit of omega w.r.t. that function. In the above example, omega is [1,1,2,3], and the H-orbit of omega is Omega2.)

[lgoettgens]

my interpretation of the documentation is that Omega and Omega2 shall be equal as sets, they just carry different actions. G acts on Omega, H acts on Omega2.
For that, we cannot take just the seeds of Omega (the points whose closure under the action of G is Omega) as the seeds of Omega2, because H can be (and in the example is) smaller.

Indeed, good catch. I opted to use Omega as an iterable of the seeds of Omega2 (with closed=true) as was already sone previously in the now replaced action_homomorphism(GSetByElements{FinGenAbGroup})).

(If we are instead interested in an H-orbit of a particular element omega of Omega, w.r.t. the action induced by phi, then this is something different. For that, we can first create the action function for H from phi and the action function of Omega, and then create the H-orbit of omega w.r.t. that function. In the above example, omega is [1,1,2,3], and the H-orbit of omega is Omega2.)

I am not quite sure which of the two scenarios is the correct one for my use-case (in the cases of non-surjective homs). But just to make sure: With the now updated code, I could compute the H-orbit of some omega by orbit(Omega2, omega), right?

[ThomasBreuer]

Concerning the idea to document induce only for the case that the map phi is a GAPGroupHomomorphism{U, T} but to allow also for a Map{U, T}, I thought that a Map between groups is always assumed to be structure preserving (i. e., a group homomorphism).

The manual is not good enough in this respect: The section "Group homomorphisms" talks only about GAPGroupHomomorphism{S, T}, and the section "Maps" belongs to "Groups"/"Abelian Groups"/"Morphisms".

[ThomasBreuer]

I am not quite sure which of the two scenarios is the correct one for my use-case (in the cases of non-surjective homs). But just to make sure: With the now updated code, I could compute the H-orbit of some omega by orbit(Omega2, omega), right?

Yes.

However, if one is interested just in one particular H-orbit, it might be cheaper to compute this directly, without creating the perhaps much too large H-set Omega2. For example, if H is the trivial group then computing the H-orbit of an element in Omega is cheap, but describing Omega as an H-set can be expensive. (We need a better way to describe Omega as an H-set, relative to the known G-set, and without the need to write down all elements.)

[ThomasBreuer]

Looks good. Thanks.

[lgoettgens]

However, if one is interested just in one particular H-orbit, it might be cheaper to compute this directly, without creating the perhaps much too large H-set Omega2. For example, if H is the trivial group then computing the H-orbit of an element in Omega is cheap, but describing Omega as an H-set can be expensive. (We need a better way to describe Omega as an H-set, relative to the known G-set, and without the need to write down all elements.)

Indeed. My idea for that would be to add a function induced_action_function(Omega, phi) that returns the fun object from this PR, and then rewrite _induce to make use of that action function.
If we export that function as well, one could make use of it as orbit(H, fun, omega) for the H-orbit of omega.
What dou you think about this idea @ThomasBreuer ?

[ThomasBreuer]

My idea for that would be to add a function induced_action_function(Omega, phi) that returns the fun object from this PR, and then rewrite _induce to make use of that action function.
If we export that function as well, one could make use of it as orbit(H, fun, omega) for the H-orbit of omega.
What do you think about this idea @ThomasBreuer ?

Yes.
We discussed yesterday that for some purposes, it is good to work with explicit G-sets, and for other purposes, it is good to work directly with their ingredients (the points, the group, the function). Thus it makes sense to switch between the two viewpoints.

[lgoettgens]

I now added induced_action and induced_action_function where the latter is just a shorthand for induced_action(action_function(Omega), phi). It is just there for convenience.

[ThomasBreuer]

Good.
(I am a bit surprised that comparing the two functions with == works.)