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We prove that the derived length of a solvable group is bounded in terms of certain invariants associated to the set of character degrees and improve some of the known bounds. We also bound the derived length of a Sylow -subgroup of a solvable group by the number of different -parts of the character degrees of the whole group.

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Given a finite solvable group , we say that has property if every set of distinct irreducible character degrees of is (setwise) relatively prime. Let be the smallest positive integer such that satisfies property . We derive a bound, which is quadratic in , for the total number of irreducible character degrees of . Three exceptional cases occur; examples are constructed which verify the sharpness of the bound in each of these special cases.

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Jiping Zhang 《代数通讯》2013,41(9):4249-4258
In this paper we determine the structure of finite solvable groups with non-connected character degree graph and improve the related known results.  相似文献   

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《Journal of Algebra》2007,307(1):300-305
Recent results of the author prove the existence of bijections of certain sets of irreducible characters which satisfy many nice properties. In particular, these correspondences imply a strengthened version of the McKay Conjecture for solvable finite groups which takes into account fields of values and Schur indices. In the present paper, we prove that these correspondences have also the additional property that the degree of the corresponding character divides the degree of the original character, and we say something about the quotient as well.  相似文献   

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In this note, we study finite groups possessing exactly one nonlinear non-faithful irreducible character. Our main result is to classify solvable groups that satisfy this property. Also, we give examples to show that these groups need not to be solvable in general.  相似文献   

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Chenggong Hao  Ping Jin   《Journal of Algebra》2008,320(12):4092-4101
Let π(G,A):IrrA(G)→Irr(CG(A)) be the Glauberman–Isaacs correspondence, where G and A are finite groups with coprime orders and A acts on G by automorphisms. Let B be a subgroup of A. In this setting, we give some new conditions for the fixed-point subgroups CG(A) and CG(B) such that χπ(G,A) is an irreducible constituent of the restriction of χπ(G,B) to CG(A) for all χIrrA(G).  相似文献   

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In this paper, the naturalness of the appearance of the group approach in the study of problems of algebraic commutation is considered. New, previously unknown solutions are obtained. Special attention is paid to an example of commutation obtained from the icosahedral equation. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 662–670, May, 1997. Translated by A. I. Shtern  相似文献   

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《Journal of Algebra》2003,259(2):353-360
We present a new method of computing the Schur index associated with an irreducible complex character of a finite group over the field of rational numbers. The method uses the Isaacs–Dade character correspondence to reduce to a subgroup of the original group.  相似文献   

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