Generating properties of Singer elements in some classical groups
by
Áron Bereczky
Rényi Institute, Budapest, Hungary

A set S of non-trivial elements in a finite group G is called an anti-generating set if for any g in G there is an s in S such that <g, s> is a proper subgroup of G. It is known that in a finite simple group every anti-generating set must contain at least two elements. We will discuss how the presence of the Singer elements in certain finite classical groups results in much stronger explicit lower bounds for the cardinalities of anti-generating sets. These results depend on a complete description of the maximal overgroups of Singer elements in the finite classical groups and on some calculations concerning the fixity ratios of certain primitive permutation representations of classical groups.

Date received: July 30, 2000