Abundant p-singular elements in finite classical groups

作者：Niemeyer Alice C; Popiel Tomasz^*; Praeger Cheryl E

来源：Journal of Algebra, 2014, 408: 189-204.

DOI：10.1016/j.jalgebra.2013.09.021

• Finite groups of Lie type
• Proportions of elements
• p-Singular elements

In 1995, Isaacs, Kantor and Spaltenstein proved that for a finite simple classical group G defined over a field with q elements, and for a prime divisor p of vertical bar G vertical bar distinct from the characteristic, the proportion of p-singular elements in G (elements with order divisible by p) is at least a constant multiple of (1-1/p)/e, where e is the order of q modulo p. Motivated by algorithmic applications, we define a subfamily of p-singular elements, called p-abundant elements, which leave invariant certain 'large' subspaces of the natural G-module. We find explicit upper and lower bounds for the proportion of p-abundant elements in G, and prove that it approaches a (positive) limiting value as the dimension of G tends to infinity. It turns out that the limiting proportion of p-abundant elements is at least a constant multiple of the Isaacs-Kantor-Spaltenstein lower bound for the proportion of all p-singular elements.

• 出版日期2014-6-15