User ben fairbairn - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:08:59Z http://mathoverflow.net/feeds/user/26255 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108978/has-any-attempt-been-made-to-classify-finite-groupoids/114666#114666 Answer by Ben Fairbairn for Has any attempt been made to classify finite groupoids? Ben Fairbairn 2012-11-27T16:05:26Z 2012-11-27T16:05:26Z <p>Asking to classify finite groups in general is essentially a pie in the sky question. We know that every finite group is 'built up' of finite simple groups, but even with those classified there are still many different ways a given set of groups can be combined to produce new ones. Worse still, enumeration of finite groups seems to suggest what one would intuitively expect to be true: the ones with less structure can be glued together in far more ways than those with rich and complicated structure.</p> <p>More explicitly, groups of order at most 2000 or so have been classified (see for instance this ten year old paper of Breach, Eick and O'Brian:</p> <p><a href="http://www.math.auckland.ac.nz/~obrien/research/2000-announce.pdf" rel="nofollow">http://www.math.auckland.ac.nz/~obrien/research/2000-announce.pdf</a> )</p> <p>The conclusion? Out of the 49 910 529 484 groups of order at most 2000 a staggering 49 487 365 422 of them have order 1024 - almost every group is not just nilpotent but is in fact a 2-group!</p> http://mathoverflow.net/questions/100446/number-of-generators-of-a-subgroup-of-a-finite-simple-group/106512#106512 Answer by Ben Fairbairn for Number of generators of a subgroup of a finite simple group Ben Fairbairn 2012-09-06T14:11:00Z 2012-09-06T14:11:00Z <p>Nobody seems to have mentioned the work of Burness, Liebeck and Shalev yet:</p> <p><a href="http://www.personal.soton.ac.uk/tb1u06/docs/maxgen26.pdf" rel="nofollow">http://www.personal.soton.ac.uk/tb1u06/docs/maxgen26.pdf</a></p> <p>They prove that if $S$ is a non-abelian finite simple group and $H$ is a maximal subgroup of $S$ then $d(H)\leq4$. Furthermore, there are infinitely many examples that attain this bound.</p>