2016/06/07 by Craven, David A
#FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1606.02326
In 1998, Liebeck and Seitz introduced a constant t(G), dependent on the root system of a reductive algebraic group G and proved that if x is a semisimple element of order greater than t(G) in G then there exists an infinite subgroup of G stabilizing the same subspaces of L(G) as x. The values for t(G) are 12, 68, 124 and 388 for G=G2,F4,E6,E7 respectively. In this paper we obtain a similar result for these groups and the minimal module Vmin, obtaining significantly smaller numbers, namely 4, 18, 27 and 75 respectively (with some small conditions on the element x that are not important for applications). Note that both t(G) and these new bounds are sharp. As a corollary we eliminate several potential maximal subgroups PSL2(q0) of these groups that seem difficult to eliminate through other means, along with other groups. This paper forms part of the author's programme to vastly reduce the number of putative maximal subgroups of exceptional groups of Lie type.