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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 |. G| 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. © 2013 Elsevier Inc.
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- 2 Finished
1/01/08 → 31/12/11
1/01/07 → 31/12/12