Add letters function for PcGroupElem

[jamesnohilly]

As a starting point for this PR, I've borrowed the syllable functions for PcGroup from PR #4108 to use as a reference for planned work.

Planned work for this PR:

• New function letters for (Sub)PcGroupElem.
• A check in _exponent_vector for whether the input is really "canonical".
• Test cases in test/Groups/pcgroups.jl for the new functions.
• Add doc strings for new functions.

Suggestions from @fingolfin

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 84.60%. Comparing base (da7acb0) to head (1b80584).
Report is 50 commits behind head on master.

Additional details and impacted files

@@           Coverage Diff            @@
##           master    #4202    +/-   ##
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  Coverage   84.59%   84.60%            
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  Files         631      640     +9     
  Lines       84899    85140   +241     
========================================
+ Hits        71824    72032   +208     
- Misses      13075    13108    +33     

Files with missing lines	Coverage Δ
src/Groups/pcgroup.jl	75.34% <100.00%> (+4.61%)	⬆️

... and 57 files with indirect coverage changes

[fingolfin]

Thanks @jamesnohilly to clarify: what you posted in the description is not (yet) what you did, but a description of the work programme, right? Also, should mention that the code here was taken from PR #4108 by @fieker and is only the starting point.

[jamesnohilly]

Yes, that's correct! I've now updated the description to make it clearer and also included a note that the code originates from PR #4108 as the starting point.

[lgoettgens]

some comments. Maybe more directed towards @fingolfin and @ThomasBreuer instead of @jamesnohilly

[lgoettgens]

why does it only work for syllables in canonical form? I would at least expect the function to test this requirement.
And I think this requirement makes it impossible to construct certain elements in certain groups. Consider < a, b | a^2 = 1, b^2 = 1, aba = bab > (Weyl group of type A2). The element aba has no representation in "syllables in canonical form" (or maybe I misunderstood what canonical form means here)

[ThomasBreuer]

The point is that there are different useful ways to encode a group element that is given by some kind of word. For general elements in finitely presented groups, there is no normal form, and an element is represented by a sequence of letters or syllables.
For elements in polycyclic groups, there is a normal form which corresponds to an exponent vector, and many low level functions deal just with these exponent vectors. In situations where one needs a word-like description of an element, it may be necessary to switch from the exponent vector to a syllable description and back. Often it is not needed to do actual computations (such as normalizing the data) in such a case. Testing whether the data are admissible can make sense; in this case, there should be a check = false option that allows one to skip these tests.

[lgoettgens]

Thanks for the explanation. I might have mixed up FP and PcGroups when thinking about this.

[jamesnohilly]

I've added this check (enabled by default) to verify if the syllables are in the required canonical form in the latest commit.

[fingolfin]

Suggested change

	end
	end

[fingolfin]

@jamesnohilly you can apply this suggestion via the web interface by using the "Commit suggestions" button. Though you then should also "fetch" the changes from git into your local branch.

[jamesnohilly]

Ah, I never noticed this, thanks! I've applied this suggestion in the latest commit.

[fingolfin]

@jamesnohilly tests are still failing. are on onto that, or do you need assistance?