We describe an algorithm to compute a composition tree for a matrix group defined over a finite field, and show how to use the associated structure to carry out computations with such groups; these include finding composition and chief series, the soluble radical, and Sylow subgroups.

Suppose that a group G acts transitively on the points of a nonDesarguesian plane, P. We prove first that the Sylow 2-subgroups of G are cyclic or generalized quaternion. We also prove that P must admit an odd order automorphism group which acts transitively on the set of points of P. 1 MSC(2000): 20B25, 51A35.

Given two finite p-local finite groups and a fusion preserving morphism between their Sylow subgroups, we study the question of extending it to a continuous map between the classifying spaces. The results depend on the construction of the wreath product of p-local finite groups which is also used to study p-local permutation representations.

In a connected group of finite Morley rank, if the Sylow 2-subgroups are finite then they are trivial. The proof involves a combination of modeltheoretic ideas with a device originating in black box group theory. Mathematics Subject Classification (2000). 03C60, 20G99.

Let G be a finite group, p a prime, and P a Sylow p-subgroup of G. Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of p′-degree of G and the irreducible characters of p′-degree of NG(P ), which preserves field of values of correspondent characters (over the p-adics). This strengthening of the McKay conjecture has ...

Let A and G be finite groups of relatively prime orders and assume that A acts on G via automorphisms. We study how certain conditions on G imply its solvability when we assume the existence of a unique A-invariant Sylow p-subgroup for p equal to 2 or 3. Mathematics Subject Classification (2010). Primary 20D20; Secondary 20D45.

Let G be a finite group and let R be a complete discrete valuation domain of characteristic 0 with residue field k of characteristic p and let S be R or k. The cohomology rings H∗(K,S) for subgroups K of G together with restriction to subgroups of G, transfer from subgroups of G and conjugation by elements of G gives H∗(−, S) the structure of a Mackey functor. Moreover, the group HSplenS(K) of ...