Saturated formations and Sylow normalisers

Abstract

Sufficient conditions are provided in order that some classes of finite soluble groups, defined by properties of the Sylow normalisers, are saturated formations.

Problems of Connectivity between the Sylow Graph,the Prime Graph and the Non-Commuting Graph of a Group

“…As noted in [58], we may introduce the group class operator N, defined by the rule: G belongs to N was largely studied in [54][55][56][57][58][59][60], but the study of properties of normalisers appear also [62][63][64][65][66][67][68][69][70][71][72]. For instance, [63,Corollary 2] shows that the class of all p-groups is N-closed and [68,Theorem 2] shows that the same is true for the class of all nilpotent groups.…”

“…coincides with the set of all prime divisors of G   G  and has been studied by Kazarin and others in [55], even if the origins go back to some works of De Vivo and others [54,[57][58][59][60], where it appears for the first time a graph with the same properties of   G  . The Sylow graph is useful for problems of formations of finite groups (see [61] for the notion of formation of finite groups) and properties of normalizers of Sylow subgroups.…”

“…For solvable groups, the answer was known earlier as corollary of results in [54,[57][58][59][60], where a different approach was adopted. [55,Main Theorem] influenced heavily [56], in which a gap occurred in the proof of the connectivity of without CFSG.…”

“…A further improvement is [64,Theorem 2], where it is considered the class of all p-nilpotent groups. In the class of all solvable groups, relations among N and classes of groups which are closed with respect to forming subgroups (briefly said S-closed), were investigated in [56][57][58][59][60]. These researches make strong use of some techniques of the theory of formations of groups, which has been largely exploited in the last years by Ballester-Bolinches, Shemetkov and Skiba (terminology and notations which we are using can be found in fact in [61,67]).…”

“…These researches make strong use of some techniques of the theory of formations of groups, which has been largely exploited in the last years by Ballester-Bolinches, Shemetkov and Skiba (terminology and notations which we are using can be found in fact in [61,67]). For instance, a significant notion is that of lattice-formation, which is quite technical to explain here (see [56][57][58][59][60]67]), but has a fundamental role in many topics of formations of finite groups. Indeed, an immediate consequence of Theorem 3.1 is:…”