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On the p-part of character degrees of solvable groups. (English) Zbl 0712.20004

Représentations linéaires des groupes finis, Proc. Colloq., Luminy/Fr. 1988, Astérisque 181-182, 217-220 (1990).
[For the entire collection see Zbl 0699.00023.]
The following theorem is proved. Theorem: Let G be solvable with \(O_ p(G)=\{1\}\) and \(p^ 2\) does not divide \(\beta\) (1) for all irreducible Brauer p-characters of G. Then: a) G has elementary abelian Sylow p- subgroups. b) \(p^ 2\) does not divide \(\chi\) (1) for all complex irreducible characters \(\chi\) of G. The proof of the theorem has been established by means of Clifford theory and theory of the Fitting subgroup of a finite group.
Reviewer: R.W.van der Waall

MSC:

20C15 Ordinary representations and characters
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D60 Arithmetic and combinatorial problems involving abstract finite groups

Citations:

Zbl 0699.00023