Finite groups with normal intersections of Sylow 2-subgroups

Abstract

Recently there have been published a number of papers concerning the influence of the intersections of the Sylow 2-subgroups of a finite group on the structure of the group itself. The first was the paper of M. Suzuki [i] on 7"7--groups, in which he described the finite groups with trivial intersections of Sylow 2-subgroups. Most of the results in this direction are connected with restrictions on the internal structure of the intersections: cyclicity, commutativity, small rank, and so on (V. D. Mazurov, S. A. Syskin, G. Stroth, M. Herzog, and others). Gomi [2] investigated the case where the intersection of any two distinct Sylow 2-subgroups P and ~ is contained in the center of some Sylow 2-subgroup.It turns out, in this situation, that Pn~ is contained in the intersection of the centers of P and ~ .Our note is devoted to a more general problem, formulated by Mazurov and Syskin in an unpublished list of unsolved problems: In which finite groups is the intersection of any two Sylow 2-subgroups normal in at least one of them? For brevity, groups satisfying this condition will be called NI*-groups.In the sequel we consider only finite groups.A group in which the intersection of any two Sylow 2-subgroups is normal in each is called a NI-group.THEOREM. Every NI*-group is a NI-group. If P is a Sylow 2-subgroup of a NI-group and O~(G)= / , then ~r~ I (P) is contained in Z(P) 9The notation is standard.In particular, br21 (P) is the subgroup generated by all elements of order 2 (involutions) in P ; Z(P) is the center of P. 02(G) is the largest normal 2-subgroup of ~ .

Sylow Subgroups of the Chevalley Groups and Associated (Weakly) Finitary Groups and Rings

“…Suzuki [18] proved that a finite (non-Abelian) simple group with independent Sylow 2-subgroups is a Chevalley group of Lie rank 1 over a field of characteristic 2. By the middle of 1970s there had been completed a classification of finite simple groups in which the intersection of every pair of Sylow 2-subgroups P and Q is normal either in P or in Q. Sylow 2-subgroups of a such group are either independent or Abelian, by [4]. Also, Kabanov, Makhnev and Starostin raised Question 5.14 in the Kourovka Notebook [5]; Questions 5.14, (c) and (d) coincide, respectively, with (A) and (B) at p = 2 and are still open.…”