problem with TestIdentityAction

[ThomasBreuer]

The following happens in the current master branch as well as in released versions
(at least 4.10.0, 4.9.1, 4.7.6).

gap> g:= GL(2,3);;
gap> SparseActionHomomorphism( g, [ [ Z(3), Z(3) ] ] );
Error, Action not well-defined. See the manual section
``Action on canonical representatives''. at /.../gapmaster/gap/lib/oprt.gd:441 called from
TestIdentityAction( acts, pnt, act 
 ) at /.../gapmaster/gap/lib/oprt.gd:840 called from
...

The function SparseActionHomomorphism is called in a generic NiceMonomorphism method for matrix groups, try for example Size( Group( GeneratorsOfGroup( GL( 2, Integers mod 6 ) ) ) ).
The arguments of the (undocumented) function TestIdentityAction are the list of group generators, the list of seed vectors, and the action function (here the default OnPoints),
and the idea behind TestIdentityAction is to check whether the One of the first generator fixes each seed vector w.r.t. the action function.
In the situations where SparseActionHomomorphism is called in GAP library code,
this does not run into an error because then the second argument is the (list of rows of) the identity element in the matrix group.

gap> g:= GL(2,3);;
gap> SparseActionHomomorphism( g, One( g ) );
<action epimorphism>

Note that in this case, TestIdentityAction tests in fact whether the One of the first generator conjugates the list of seed vectors to itself, in the sense that <pnt>^<one> is computed as <one> * <pnt> * <one>, which is fine only if <pnt> is square.
If one specifies the action function OnRight instead of OnPoints then the above example works.
(There is even a manual example that shows this situation.)

gap> g:= GL(2,3);;
gap> SparseActionHomomorphism( g, [ [ Z(3), Z(3) ] ], OnRight );
<action epimorphism>

I think that the check in TestIdentityAction must apply the identity element to each seed vector separately, since we cannot assume that the action function knows how to deal with a list of seed vectors.
Perhaps the cleanest solution would be to replace the TestIdentityAction call by an appropriate Assert call; note that TestIdentityAction seems to be not used in other places.
The only argument against this change is that currently the list of seed vectors gets replaced by the result of TestIdentityAction (why?), but on the other hand it seems to be allowed to skip TestIdentityAction via the option NoTestAction.

[hulpke]

As far as I can see the problem is with declaring SparseActionHomomorphism through OrbitishFO (Orbit-style operations take one seed only).
This must have been a bug all the time, and just never showed up.
I have submitted a PR that splits this up (and then calls TestIdentityAction separately on each of the seeds).

As for calling TestIdentityAction in the first place, this is (prompted by bug reports) to catch issues, such as Orbit(G,[1,3,2],OnSets), in which the point does not satisfy implicit assumptions on the action.

[olexandr-konovalov]

I removed "backport-to-4.10" label from this issue - one can not really backport an issue. But one can backport a pull request, and I have added appropriate labels to #3281.