Conjugate \(\pi\)-normally embedded Fitting functors
This paper continues the study of Fitting functors on the class \({\mathcal S}\) of all finite soluble groups, begun by \textit{J. C. Beidleman}, \textit{B. Brewster}, \textit{P. Hauck} [Math. Z. 182, 359-384 (1983; Zbl 0518.20014), Math. Proc. Camb. Philos. Soc. 101, 37-55 (1987; Zbl 0632.20013)]. A Fitting functor is a map f assigning to each \(G\in {\mathcal S}^ a \)non- empty set f(G) of subgroups of G such that (with slight abuse of notation) \(\mu (f(N))=\mu (N)\cap f(G)\) for every normal embedding \(\mu\) : \(N\to G\); motivating examples for Fitting functors are \({\mathcal F}\)- injectors and \({\mathcal F}\)-radicals for Fitting classes \({\mathcal F}\). It is called conjugate if f(G) is always a conjugacy class of subgroups of G, and \(\pi\)-normally embedded, \(\pi\) a set of prime numbers, if f(G) always consists of \(\pi\)-normally embedded subgroups of G; here a subgroup H of \(G\in {\mathcal S}\) is called \(\pi\)-normally embedded in G if any Hall \(\pi\)- subgroup K of H is a Hall \(\pi\)-subgroup of its normal closure in G. The main aim of the present paper is to give a description and various properties of conjugate \(\pi\)-normally embedded Fitting functors, and to investigate I-normally embedded Fitting functors, where I is the index set of a partition of the set of all primes (I-normality being defined via \(\pi_ i\)-normality for all \(i\in I\) in the obvious way); the case when I corresponds to the partition of the set of all primes into 1- element sets yields the normally embedded Fitting functors which were dealt with in the above-mentioned papers. Finally, Fitting classes with \(\pi\)-normally embedded injectors are studied.
conjugate \(\pi\)-normally embedded Fitting functors
scientific article; zbMATH DE number 4091734
scientific article