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Irish Mathematical Society Bulletin
Article . 1988 . Peer-reviewed
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Minimal Fitting Classes

Minimal Fitting classes
Authors: McCann, Brendan;

Minimal Fitting Classes

Abstract

These two papers [see also the following item Zbl 0667.20016] are both about Fitting Class theory. A class \({\mathcal F}\) of finite groups is called a Fitting Class if it satisfies two properties, i) if \(G\in {\mathcal F}\) and H is a normal subgroup of G then \(H\in {\mathcal F}\) and ii) if G is a finite group and H and K are both normal subgroups of G both in \({\mathcal F}\) then KH is in \({\mathcal F}\). It is not difficult to see that the intersection of Fitting Classes is a Fitting Class, so given a set of finite groups one can talk about the Fitting Class generated by them. The particular problem that concerns both of these papers is to try to describe the Fitting class generated by a given finite group, this will be denoted by Fit(G). Since the reviewer showed that the class generated by the symmetric group of order 3 did not contain the dihedral group of order 18 [\textit{A. R. Camina}: Math. Z. 136, 351-352 (1974; Zbl 0268.20012)] little progress had been made. B. McCann made significant progress in his thesis [Univ. Würzburg (1985; Zbl 0584.20009)] by showing how to construct new Fitting Classes related to a given group. However these constructions only work for groups with a very special structure. Significantly these constructions do not apply to the symmetric group of degree three. The paper being reviewed by McCann is a short survey discussing the results above which appeared in his thesis and in an earlier paper [Arch. Math. 49, 179-186 (1987; Zbl 0601.20016)]. The paper by Bryce, Cossey and Ormerod is a more substantial work which develops the ideas from McCann's thesis. The main objective is to take the ideas of McCann and show how to relate them to the earlier work of \textit{R. Dark} [Math. Z. 127, 145-156 (1972; Zbl 0226.20013)]. They take a construction due to Dark, show how to generalize it and then relate it to the construction of McCann. They give complete proofs of all the results which makes it a very useful paper and then apply the results to \(Fit(S_ 4)\), where \(S_ 4\) is the symmetric group on 4 letters. The proofs are highly technical as are the statements of the theorems which makes them difficult to review effectively. Theorem 8.1 is description of \(Fit(S_ 4)\) which reduces the problem to a description of \(Fit(A_ 4)\). However the methods developed by McCann and extended in this paper do not make any contribution to the understanding of \(Fit(A_ 4)\) or \(Fit(S_ 3)\). The difficulty is that both these groups are metanilpotent and they do not have enough structure to enable the theory to be applied.

Keywords

intersection of Fitting Classes, Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure, Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks, finite groups, Fitting class generated by a given finite group

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
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