add RECOG.FindSL2inSL4: Find SL(2,q) as stingray subgroup of SL(4,q)

[Till-Eisen]

This PR adds gap/projective/SL4_natural.gi, which implements RECOG.FindSL2inSL4(G, N).

Given G = SL(4,q) (q odd), the function finds a subgroup U isomorphic to SL(2,q) stingray-embedded in G: concretely, a base change matrix bas is computed such that every element of U is block-diagonal diag(A, I_2) in the new basis.

[codecov]

Codecov Report

❌ Patch coverage is 91.93548% with 5 lines in your changes missing coverage. Please review.
✅ Project coverage is 86.59%. Comparing base (858e2e7) to head (6a23d81).
⚠️ Report is 9 commits behind head on master.

Files with missing lines	Patch %	Lines
gap/projective/sl4_natural.gi	91.93%	5 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master     #504      +/-   ##
==========================================
- Coverage   88.46%   86.59%   -1.87%     
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  Files          45       46       +1     
  Lines       19019    19324     +305     
==========================================
- Hits        16825    16734      -91     
- Misses       2194     2590     +396     

Files with missing lines	Coverage Δ
gap/projective/sl4_natural.gi	91.93% <91.93%> (ø)

... and 18 files with indirect coverage changes

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[fingolfin]

Good, now that your functions gets loaded, have you tried actually using it, i.e., by modifying RECOG.SLn_constructsl2 to call you code, instead?

[Till-Eisen]

This function works fine. However the functions RECOG.SLn_exceptionalgodown and RECOG.SLn_godownfromd which, given a SL(d,q) as a subgroup of SL(dim, q) (in natural representation) provide a subgroup isomorphic to SL(2,q), together they cover all possible parameter q, d and dim and have a lower running time than the function RECOG.FindSL2inSL4_natural implemented in this PR.
Thereby either the implementation lacks efficiency or the underlying algorithm is Inferior.

In the following a small benchmark. Note that the function RECOG.FindStdGens_SL uses RECOG.SLn_godownfromd for $q>10$ and RECOG.FindStdGensUsingBSGS else.

gap> d := 4;;
gap> q := 5;;
gap> g := SL(d,q);;
gap> for i in [1..1000] do RECOG.FindSL2inSL4_natural(g,20
0);; od; time;
9379
gap> for i in [1..1000] do RECOG.FindStdGensUsingBSGS(g,Ge
neratorsOfGroup(g),false, false);; od; time;
2340
q := 13;;
gap> g := SL(d,q);;
gap> for i in [1..100] do RECOG.FindSL2inSL4_natural(g,200
);; od; time;
974
gap> for i in [1..100] do RECOG.SLn_godownfromd(g,q,d,d); 
od; time;
200