Finite groups with special Sylow 2-subgroups

Let T be a Sylow 2-subgroup of a simple group PSU (3, 2ⁿ), and Z a proper subgroup belonging to the center of T. We shall prove that a simple finite group whose Sylow 2-subgroup is isomorphic to T/Z coincides with PSU (3, 2ⁿ). As a consequence we list simple groups that can be represented in the form of a product of two Schmidt groups, i.e., of minimal nonnilpotent groups.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 217–222, August, 1973.     

Sylow 2-subgroups of rational solvable groups

A long-standing conjecture proposes that a Sylow 2-subgroup S of a finite rational group must be rational. In this paper we provide a counterexample to this conjecture, but we show that if G is solvable and S has nilpotence class 2, then S actually is rational.     

Finite simple groups whose Sylow 2-subgroups possess an extra-special subgroup of index 2

Let G be a finite simple group with Sylow 2-subgroup T. If there is an extra-special sub-group of index 2 in T, then G is isomorphic to one of the following groups: $$\mathop {M_{11,} }\limits_{U_3 (q),}$$ for an appropriate odd q.     

Sylow 2-subgroups of the periodic groups saturated with finite simple groups

We prove that every 2-subgroup of a periodic group saturated with groups of Lie type over fields of odd characteristics whose Lie ranks are bounded as a whole is Chernikov. In particular, every such group is locally finite.     

Simple modules for groups with abelian Sylow 2-subgroups are algebraic

An algebraic module is a KG-module that satisfies a polynomial with integer coefficients, with addition and multiplication given by direct sum and tensor product. In this article we prove that if G is a group with abelian Sylow 2-subgroups and K is a field of characteristic 2, then every simple KG-module is algebraic.     

Finite groups in which the normalizers of pairwise intersections of Sylow 2-subgroups have odd indices

All finite semisimple groups are described in which the normalizers of pairwise intersections of Sylow 2-subgroups have odd indices; thereby, Problem 5.14(v) from the Kourovka Notebook is solved in the main.