add schur_multiplier

[ThomasBreuer]

and abelian_invariants and abelian_invariants_multiplier

resolves #2962

[thofma]

Thanks! Why not abelian_invariants_schur_multiplier (for consistency)? I have not strong opinion.

[codecov]

Codecov Report

Merging #2965 (bab90e7) into master (103262f) will decrease coverage by 0.22%.
Report is 4 commits behind head on master.
The diff coverage is 91.17%.

@@            Coverage Diff             @@
##           master    #2965      +/-   ##
==========================================
- Coverage   80.36%   80.14%   -0.22%     
==========================================
  Files         470      472       +2     
  Lines       66662    67094     +432     
==========================================
+ Hits        53573    53775     +202     
- Misses      13089    13319     +230     

Files	Coverage Δ
src/Groups/sub.jl	97.19% <100.00%> (+0.20%)	⬆️
src/Groups/GrpAb.jl	90.39% <86.36%> (-0.43%)	⬇️

... and 10 files with indirect coverage changes

[ThomasBreuer]

@thofma I have adjusted the name.

[thofma]

Thanks, looks good to me. Not sure if @fingolfin wants to have a look too?

[fingolfin]

add a comment maybe that explains in 1-2 lines what this is about? My guess: v is supposed to be a vector of integers. And this function converts between the two different conventions for what "abelian invariants" are:

1. elementary divisors, so prime powers or zero -- this is what GAP uses to represent the "abelian invariants" of a group
2. invariant factors -- a list of integers $d_1 \mid d_2 \mid \dots \mid d_k$

I linked to Wikipedia here, but e.g. Encyclopedia of Mathematics agrees as does Bourbaki. Wikipedia also cites Lang's "Algebra" in support, but he actually only states an "Elementary Divisors Theorem", but does not say what an "elementary divisor" is -- and his theorem produces what above is called "invariant factors"

But not every does, e.g. Hecke has elementary_divisors return what I call "invariant factors", and I just read a paper by Frank Lübeck where he also used this convention.

So as usual, Math is inconsistent as hell... sigh

Anyway, back to this function: it seems to convert from format 2 to format 1 above, a comment should say this.

We now have multiple choices: use format 1 or format 2; and call the result by one or by the other name... Whatever we do, we should make explicit what we mean, given that the terms alone are ambiguous.

Anyway, I am fine with sticking to either naming convention; and also , as long as we make it clear in the documentation which we use (offering a helper function to convert between them of course also would be nice)
As far as I can tell, Magma has two functions, PrimaryAbelianInvariants and AbelianInvariants which return the first resp. the second kind of sequence.

[thofma]

I noticed this math inconsistency a while ago, but I am happy to make our system consistent. It does not need to happen in this PR.

[ThomasBreuer]

O.k., I will add some comments as suggested, and leave it to the discussion of issue #2968 how to change names of functions if necessary.