Finite groups with a given set of character degrees. (English) Zbl 1348.20007
In the summary one finds: “Let \(G\) be a finite group with \(\{1,q,r,qm,q^2m,rqm\}\) as the character degree set, where \(r\) and \(q\) are distinct primes and \(m>1\) is an integer not divisible by \(q\) and \(r\). It is shown that \(G\) is solvable and the derived length of \(G\) equals 3.”
Several papers published earlier in time dealt with complex character degree sets, like \(\{1,p,q,r,pq,pr\}\) where \(p\), \(q\), \(r\) are distinct primes, and \(\{1,p\}\), and \(\{1,q,r\}\), etc.. The rich reference list gives (for instance), a paper by K. Aziziheris [Algebr. Represent. Theory 14, No. 5, 949-958 (2011; Zbl 1245.20004)], by M. L. Lewis [J. Algebra 206, No. 1, 235-260 (1998; Zbl 0915.20003)] and by T. Noritzsch [J. Algebra 175, No. 3, 767-798 (1995; Zbl 0839.20014)].
In order to obtain their results, the authors develop a lot of theory to reach their goals. We kindly refer the interested reader to the details of the contents. Knowledge of cyclotomic polynomials, Lie-group representations, Steinberg characters, do play a rôle as well as the classification of the finite simple groups.
Several papers published earlier in time dealt with complex character degree sets, like \(\{1,p,q,r,pq,pr\}\) where \(p\), \(q\), \(r\) are distinct primes, and \(\{1,p\}\), and \(\{1,q,r\}\), etc.. The rich reference list gives (for instance), a paper by K. Aziziheris [Algebr. Represent. Theory 14, No. 5, 949-958 (2011; Zbl 1245.20004)], by M. L. Lewis [J. Algebra 206, No. 1, 235-260 (1998; Zbl 0915.20003)] and by T. Noritzsch [J. Algebra 175, No. 3, 767-798 (1995; Zbl 0839.20014)].
In order to obtain their results, the authors develop a lot of theory to reach their goals. We kindly refer the interested reader to the details of the contents. Knowledge of cyclotomic polynomials, Lie-group representations, Steinberg characters, do play a rôle as well as the classification of the finite simple groups.
Reviewer: Robert W. van der Waall (Amsterdam)
MSC:
| 20C15 | Ordinary representations and characters |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
Keywords:
finite solvable groups; character degrees; derived lengths; Taketa inequality; character degree sets; irreducible complex charactersReferences:
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