Frattiniclasses and maximal subclasses, in particular of saturated formations of finite groups
type: Item
instance of: scholarly article
title: Frattiniclasses and maximal subclasses, in particular of saturated formations of finite groups
zbMATH Open document ID: 0652.20025
DOI: 10.1016/0021-8693(88)90242-6
author: Ute Christina Herzfeld
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: This paper collects some simple observations, taken from the second version of the author's Ph. D. thesis [Mainz, 1986; Zbl 0602.20022], concerning Frattini classes of classes of finite groups. For any closure operator A on classes of finite groups (in the sense of \textit{P. Hall} [Proc. Camb. Philos. Soc. 59, 531-553 (1963; Zbl 0118.036)]) the A- Frattini class \(\Phi_ A({\mathcal X})\) of an A-closed class \({\mathcal X}\) is the class of all \(G\in {\mathcal X}\) such that \({\mathcal X}=A{\mathcal Y}\) whenever \({\mathcal Y}\subseteq {\mathcal X}=A({\mathcal Y}\cup \{G\})\); for finitary A, it can be characterised as being the intersection of all maximal A-closed subclasses of \({\mathcal X}\), if such exist, and \({\mathcal X}\), otherwise. The non-technical result of the paper is the following straightforward Theorem: If \({\mathcal X}\) is a local formation of finite groups minimally locally defined by the formation function f, and if for all primes p in the characteristic of \({\mathcal X}\), f(p) is a local formation containing a non-trivial group, then \(\Phi_ A({\mathcal X})={\mathcal X}\), that is, \({\mathcal X}\) has no maximal local subformation.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20F17
zbMATH DE Number: 4063363
zbMATH Keywords: Frattini classes
zbMATH Keywords: classes of finite groups
zbMATH Keywords: local formation of finite groups
zbMATH Keywords: maximal local subformation
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90242-6
OpenAlex ID: W1995965656
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description: scientific article; zbMATH DE number 4063363