Permutability of injectors with a central socle in a finite solvable group. (English) Zbl 1370.20014
The authors answer an open question of K. Doerk and T. Hawkes [Finite soluble groups. Berlin etc.: W. de Gruyter (1992; Zbl 0753.20001). Namely, it is shown that if \({\mathcal Z}^\pi\) is the Fitting class formed by the finite solvable groups whose \(\pi\)-socle is central (where \(\pi\) is a set of prime numbers), then the \({\mathcal Z}^\pi\)-injectors of a finite solvable group \(G\) permute with the members of a Sylow basis in \(G\).
To be specific, a Sylow basis \(\Sigma\) in \(G\) is defined as a set of Sylow subgroups of \(G\) with \(|\Sigma\cap \text{Syl}_p(G)|=1\) for each prime number \(p\) such that all pairs of members of \(\Sigma\) permute with each other. In this paper, the authors prove as their main theorem that the \({\mathcal Z}^\pi\)-injectors of \(G\) are system permutable in \(G\), thereby indeed answering the above question. In order to do so, they have to develop a lot of properties connected to representation theory of certain particular groups centering around extra-special groups, which was done very intricate and detailed, but beautifully.
The paper is organized, said here in a condensed way, as follows:
1. Introduction; 2. Quoted results (Not only results to be used are mentioned, but technical lemmas needed later have been proved, too.); 3. Extra-special irreducible groups (Needed is here the notion of extra-special irreducible graph. That is, an extra special \(r\)-graph \(R\) (\(r\) a prime number) admits a finite graph \(Q\) acting on \(R\) in such a way, that \([R,Q]= R\)[(center of \(R),Q]= \{1\}\) and there exists no extra-special subgroup \(V\) satisfying (center of \(R)< V< R\) with \(V\) normal in \(QR\)); 4. Minimal groups with an injector which is not system permutable (Consequences of the hypothesis in the title of this Section 4 have been stated and proved.); 5. Minimal groups with a \({\mathcal Z}^\pi\)-injector which is not system permutable (Results derived in earlier sections are used here to establish particular structures of groups; representation theory of modules also appears) 6. and 7. (we omit the titles in this review). (Here, former obtained results are connected to each other in order to obtain special configurations about representation theory of groups in which extra-special irreducible groups are involved.) 8. Proof of the theorem (i.e., the result mentioned in the beginning of this review).
The whole thirty-five pages of work as done in this paper must have taken a lot of time to work out. We close by mentioning some literature involved [the authors, J. Group Theory 12, No. 4, 511–538 (2009; Zbl 1200.20013); ibid. 333, No. 1, 139–160 (2011; Zbl 1248.20015); ibid. 381, 209–232 (2013; Zbl 1323.20015); Adv. Group Theory Appl. 2, 31–65 (2016; Zbl 1381.20023); Doerk and Hawkes, loc. cit.].
To be specific, a Sylow basis \(\Sigma\) in \(G\) is defined as a set of Sylow subgroups of \(G\) with \(|\Sigma\cap \text{Syl}_p(G)|=1\) for each prime number \(p\) such that all pairs of members of \(\Sigma\) permute with each other. In this paper, the authors prove as their main theorem that the \({\mathcal Z}^\pi\)-injectors of \(G\) are system permutable in \(G\), thereby indeed answering the above question. In order to do so, they have to develop a lot of properties connected to representation theory of certain particular groups centering around extra-special groups, which was done very intricate and detailed, but beautifully.
The paper is organized, said here in a condensed way, as follows:
1. Introduction; 2. Quoted results (Not only results to be used are mentioned, but technical lemmas needed later have been proved, too.); 3. Extra-special irreducible groups (Needed is here the notion of extra-special irreducible graph. That is, an extra special \(r\)-graph \(R\) (\(r\) a prime number) admits a finite graph \(Q\) acting on \(R\) in such a way, that \([R,Q]= R\)[(center of \(R),Q]= \{1\}\) and there exists no extra-special subgroup \(V\) satisfying (center of \(R)< V< R\) with \(V\) normal in \(QR\)); 4. Minimal groups with an injector which is not system permutable (Consequences of the hypothesis in the title of this Section 4 have been stated and proved.); 5. Minimal groups with a \({\mathcal Z}^\pi\)-injector which is not system permutable (Results derived in earlier sections are used here to establish particular structures of groups; representation theory of modules also appears) 6. and 7. (we omit the titles in this review). (Here, former obtained results are connected to each other in order to obtain special configurations about representation theory of groups in which extra-special irreducible groups are involved.) 8. Proof of the theorem (i.e., the result mentioned in the beginning of this review).
The whole thirty-five pages of work as done in this paper must have taken a lot of time to work out. We close by mentioning some literature involved [the authors, J. Group Theory 12, No. 4, 511–538 (2009; Zbl 1200.20013); ibid. 333, No. 1, 139–160 (2011; Zbl 1248.20015); ibid. 381, 209–232 (2013; Zbl 1323.20015); Adv. Group Theory Appl. 2, 31–65 (2016; Zbl 1381.20023); Doerk and Hawkes, loc. cit.].
Reviewer: Robert W. van der Waall (Amsterdam)
MSC:
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 20D25 | Special subgroups (Frattini, Fitting, etc.) |
| 20F14 | Derived series, central series, and generalizations for groups |
| 20C15 | Ordinary representations and characters |
Keywords:
finite solvable groups; Fitting class; extra-special \(p\)-groups; injectors; central socle; Sylow basis; representation theory of finite groups.References:
| [1] | Dark, R.; Feldman, A.; Pérez-Ramos, M. D., Persistent characterizations of injectors in finite solvable groups, J. Group Theory, 12, 511-538 (2009) · Zbl 1200.20013 |
| [2] | Dark, R.; Feldman, A.; Pérez-Ramos, M. D., Injectors with a normal complement in a finite solvable group, J. Algebra, 333, 139-160 (2011) · Zbl 1248.20015 |
| [3] | Dark, R.; Feldman, A.; Pérez-Ramos, M. D., Injectors with a central socle in a finite solvable group, J. Algebra, 381, 209-232 (2013) · Zbl 1323.20015 |
| [4] | Dark, R.; Feldman, A.; Pérez-Ramos, M. D., On extraspecially irreducible groups, Adv. Group Theory Appl., 2, 31-65 (2016) · Zbl 1381.20023 |
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| [7] | Huppert, B., Endliche Gruppen I, Grundlehren der mathematischen Wissenschaften, vol. 134 (1967), Springer-Verlag · Zbl 0217.07201 |
| [8] | Huppert, B.; Blackburn, N., Finite Groups II, Grundlehren der mathematischen Wissenschaften, vol. 242 (1982), Springer-Verlag · Zbl 0477.20001 |
| [9] | Isaacs, I. M., Character Theory of Finite Groups (1976), Academic Press, AMS Chelsea, 2006 · Zbl 0337.20005 |
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