Corrected word_problem parsing and added algorithm option

[EigenVector22]

This replaces the fragile string parsing in PermutationGroupElement.word_problem with a stable method using GAP's syllable functions (NumberSyllables, GeneratorSyllable, ExponentSyllable). This resolves parsing errors involving parentheses and generator name conflicts (e.g., x10 vs x1).

Introduces an 'algorithm' parameter:

• 'syllable' (Now default): Returns 1-based generator indices when as_list=True.
• 'legacy': Old implementation, preserved for testing but deprecated due to known bugs. Returns generator strings when as_list=True.

Updated docstrings and examples accordingly.

Fixes #36419
Also #37042 is taken care of
Related: #28556

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

[EigenVector22]

Hello, @dimpase @fchapoton

Can you please review this, and give feedback about the fix.
Thanks,

[dimpase]

there is a CI error:
https://github.com/sagemath/sage/actions/runs/14145043829/job/39633426770#step:16:1

[EigenVector22]

made the necessary changes,

[EigenVector22]

It looks like some CI checks are failing in unrelated files (linear_code.py, polynomial_element.pyx) with timeouts/cysignal errors, and there's also a Docker image pull error. These seem unrelated to the changes in permgroup_element.pyx, as local tests for that file pass?

[EigenVector22]

made the change

[EigenVector22]

Why are some tests failing? Have I done any mistake in the implementation?

[EigenVector22]

@fchapoton @dimpase, any further suggestions for revisions?

[dimpase]

[sagemath_doc_pdf-none] [spkg-install] [groups   ] docstring of sage.groups.perm_gps.permgroup_element.PermutationGroupElement.word_problem:31: WARNING: Block quote ends without a blank line; unexpected unindent. [docutils]

docs don't build - please fix

[EigenVector22]

@dimpase fixed that

[EigenVector22]

@dimpase in the failing doctest, the sequences are different but they are effectively representing the same group element, should i change the doctest?

[dimpase]

@dimpase in the failing doctest, the sequences are different but they are effectively representing the same group element, should i change the doctest?

Which one, can you point its location in the Sage file?
ideally, the doctest should be insensitive to this kind of unpredictable noise.

E.g. if the output is [1,3,2] but [1,2,3], or any other order, is equally valid, and sometimes happens, you might want to explicitly sort the output before comparing.

[EigenVector22]

okay, just wanted to confirm any other feedback, or does it look good?