1. Maximal subgroups. I will assume in this section that the finite simple groups can be classified and even that their irreducible representations in characteristicp can someday be determined. What I wish to demonstrate is that this will carry us a long way toward the determination of their maximal subgroups. It should not come as a total surprise that such a problem is tractible, since the co...

I would like to discuss two results about finite groups: (0) All finite groups of odd order are solvable. (F) A finite group is nilpotent if it admits a fixed point free automorphism of prime order. Walter Feit and I proved (0) after a prolonged joint effort [5]. A critical special case of (0) was proved by Walter Feit, Marshall Hall, Jr. and me [4]. A slightly stronger result than (F) is in th...

Fix a prime number . Graphs of isogenies of degree a power of are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called l-isogenies, resolving that, in arbitrary dimension, their structure is similar, but not identical, to the “volcanoes” occurri...

for absolute constants C6, C7. This theorem (almost) follows from [EV, Theorem 1.1], the only point being to control the dependence of implicit constants on the degree of the number field. We refer to [EV] for further information and for some motivational comments about the method. In the proof C1, C2, . . . will denote certain absolute constants. A.2. Let K be an extension of Q of degree d ≥ 2...

The orthogonal simple group 0 (3) has three conjugacy classes of maximal subgroups the form 36:L4(3). These groups are all isomorphic to each other and order 4421589120 with index 1120 in (3). In this paper, we will compute ordinary carácter table one these using technique Fischer-Clifford matrices. This is very efficient character an extension Ḡ = N.G especially where normal subgroup N element...

If p is a prime number, the poset of all nontrivial elementary abelian p subgroups of a finite group plays an important role in both group theory and representation theory. It was studied by Quillen [Qu], who proved among many other things that it is homotopy equivalent to the poset of all nontrivial p-subgroups. In the case of a p -group P , one might believe that this poset has no interest si...