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extended frattini dual

Proceedings of the Edinburgh Mathematical Society (1)

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d-FRATTINI CHIEF FACTORS
H. Bechtell
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Proceedings of the Edinburgh Mathematical Society / Volume 51 / Issue 1 / February 2008

Published online by Cambridge University Press:

04 February 2008, pp. 1-9
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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.