Weak closure and Oliver's p-group conjecture

To date almost all verifications of Oliver's p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3^n for n = 49 which refutes the weakly closed conjecture but satisfies Oliver's conjecture.Comment: 9 page

Simple connectedness of the 3-local geometry of the Monster

We consider the 3-local geometry M of the Monster group M introduced in [BF] as a locally dual polar space of the group\Omega \Gamma 8 (3) and independently in [RS] in the context of minimal p- local parabolic geometries for sporadic simple groups. More recently the geometry appeared implicitly in [DM] within the Z3-orbifold construction of the Moonshine module V " . In this paper we prove the simple connectedness of M. This result makes unnecessary the refereeing to the classification of finite simple groups in the Z3 -orbifold construction of V " and realizes an important step in the classification of the flag-transitive c-extensions of the classical dual polar spaces (cf. [Yo]). We make use of the simple connectedness results for the 2-local geometry of M [Iv1] and for a subgeometry in M which is the 3-local geometry of the Fischer group M(24) [IS]. 1 Introduction The Monster group M acts flag-transitively on a diagram geometry M which is described by the following diagram..

On the generalized Fitting group of locally finite, finitary groups

Let G be a locally finite, finitary group and F (G) the group generated by the Hirsch-Plotkin radical of G and the components of G. Our main theorem asserts that CG (F (G)) F (G). 1 Introduction The main purpose of this paper is to extend the concept of the generalized Fitting group from finite groups to locally finite, finitary groups. Recall that a group G is called locally finite if every finite subset of G lies in a finite subgroup. G is called finitary if there exist a field K and a faithful KG-module V so that [V; g] is finite dimensional for all g 2 G. G is quasi-simple if it is perfect, and G=Z(G) is simple. A component of G is a non-trivial quasi-simple, subnormal subgroup of G. The layer E(G) is defined as the group generated by the components of G. It is easy to see that distinct components of a group commute. Thus E(G)=Z(E(G)) is semisimple, that is the ( restricted) direct product of simple groups. Assume for the moment that G is finite. Then the Fitting group..

Baumann-components of finite groups of characteristic p, general theory

In this paper we introduce a new conjugacy class Bau_p(G) of p-subgroups of finite groups G of characteristic p. We then prove some factorization and decomposition theorems related to this conjugacy class. In particular, these results show that the only obstructions for B 08Bau_p(G) being normal in G are the Baumann components of G, a class of subnormal subgroups E with E/Op(E) quasisimple or SL2(p)\u2032 for p 643

General offender theory

We present an offender theory that is symmetric in offender and offended group and also a replacement theorem that does not need that the groups in question are abelian. We then use this theory to define variations of Thompson and Baumann subgroups and prove a general Baumann argument. (C) 2017 Elsevier Inc. All rights reserved