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d-FRATTINI CHIEF FACTORS

Published online by Cambridge University Press:  04 February 2008

H. Bechtell
H. Bechtell
Affiliation:
Mathematics Department, University of New Hampshire, Durham, NH 03824, USA ([email protected])
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Abstract

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The concepts of d- and nd-Frattini chief factors of a finite group are introduced. Their ingrainment into that of the extended Frattini dual subgroup becomes the natural dual to Frattini and supplemented chief factors. Not only does a dual of the strengthened form of the Jordan–Hölder theorem arise, but also the $p$-nilpotent radical becomes the intersection of the centralizers of the nd-Frattini chief factors. As a result, a class $\mathfrak{F}$ of groups is a full integrated local formation $\mathrm{LF}(f)$ if and only if each nd-Frattini chief factor in $G\in\mathfrak{F}$ is $f$-central.

Keywords

Primary 20D25Secondary 20D10extended Frattini duald-Frattini chief factorssaturated formations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 51 , Issue 1 , February 2008 , pp. 1 - 9
Copyright
Copyright © Edinburgh Mathematical Society 2008