Catalogue of groups with short finite presentations

Question

For various types of groups, there exist catalogues of those groups of the particular type which are "small" in a certain sense. — For example:

• The GAP Small Groups Library catalogizes groups of small order,

• The GAP Transitive Groups Library and the GAP Primitive Groups Library catalogize transitive, respectively primitive, subgroups of symmetric groups of small degree,

• The GAP Perfect Groups Library catalogizes perfect groups of small order,

and so on.

Question: Does there exist such data library for groups with "short" finite presentations?

Remarks:

• There is a 12-years-old question asking for such database. At that time, apparently such database was not available. But maybe this has changed in the meantime(?)

• When setting the length of a finite presentation equal to the sum of the lengths of the relators, for every presentation on two generators of length at most $10$, it is straightforward to decide whether the corresponding group $G$ is trivial, nontrivial but finite, or infinite (if $G$ is infinite, this can be seen from the abelian invariants of $G$, $G'$ or $G''$, and if $G$ is finite, coset enumeration finishes even with very small limits). Where things start to get more interesting is length $11$. One of the few examples for which deciding finiteness seems more tricky is the group $$ G \ := \ \langle a, b \ | \ a^3 = ab^{-3}a^{-1}b^{-1}a^{-1}b = 1 \rangle. $$ What is easy to see is that $G/G'' \cong {\rm C}_{37} \rtimes {\rm C}_9$, and that $G''$ is perfect — but beyond that, things seem to get more difficult. Almost for sure, people have considered this presentation (and other similar presentations) before. Looking into an appropriate data library would tell what is known about that (and other similar) groups immediately.

Answer 1

I would very much like to have such a database and would like to contribute to its development. Prompted by this question, we talked about what such a database could look like (e.g. in terms of groups covered, functionality etc.) at a discussion session of a workshop in Manchester with Ian Leary, Marco Linton, Saul Schleimer and Henry Wilton. I encourage anyone interested in this to contact me.

As some inspiration, I'll link this mathoverflow question on atlas-like websites.

Answer 2

The example group given has order 333. The automata program in the Monoid Automata Factory, when given this input:

#aaa,ab'b'b'a'b'a'b
_RWS := rec
(
  isRWS := true,
  ordering := "shortlex",
  generatorOrder := [a,A,b,B],
  inverses := [A,a,B,b],
  equations := 
  [
   [a*a*a,IdWord],
   [a*B*B*B*A*B*A*b,IdWord]
  ]
);

runs for about six hours but confirms this.

Separately, if you run the above input through the kbprog program from kbmag, with the option -t 1000000 (else it runs for days), it finds a confluent rewriting system after about six minutes; taking one of the last additional relations found in that rewriting system (such as "[B^2abAB,b^2Ab^2]" from the .kbprog output file) gives you this GAP input file:

f := FreeGroup("a","b");;
g := f / [ f.1*f.1*f.1,f.1*f.2^-1*f.2^-1*f.2^-1*f.1^-1*f.2^-1*f.1^-1*f.2,f.2^-1*f.2^-1*f.1*f.2*f.1^-1*f.2^-1*f.2^-1*f.2^-1*f.1*f.2^-1*f.2^-1 ];
Size(g);

which prints 333.

Thus two completely independent Knuth-Bendix implementations show this is a finite group of order 333.