Derek Holt

Enumerating transitive groups and bounding generator numbers

Computer databases that contain information on various types of groups can
play a vital role in research. Examples include finite groups up to order 2000,
primitive permutation groups up to degree 4095, and transitive permutation
groups, recently extended to degree 48.

In the first part of the talk, we provide some details on the recent
successful lengthy computer calculations involved in the enumeration of the
195,826,352 transitive groups of degree 48 (i.e. conjugacy classes in the
symmetric group).

For a finitely generated group G, let d(G) be the smallest number of elements
required to generate G.

In the second part of the talk, we survey results bounding d(G) for various
types of finite permutation and matrix groups of a given degree. In particular,
Lucchini, Menegazzo and Morigi proved in 2000 that, for transitive permutation
groups of degree n, we have d(G) <= c n/sqrt{log n} for some unspecified constant c. In his PhD thesis in 2014, Gareth Tracey proved that, except for finitely many exceptional values of n, we can take c = sqrt{3}/2 = 0.866... (This bound is exact when n = 8, but it is conjectured that the best possible bound for sufficiently large n is about half of this.) By explicitly computing d(G) for the transitive groups of degree 48, Gareth has recently succeeded in improving this result by eliminating the exceptional cases: the result holds in general with c = sqrt{3}/2.