944 research outputs found

    Characters of p'-degree and Thompson's character degree theorem

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    A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group GG is 1 or divisible by a prime pp, then GG has a normal pp-complement. We obtain a significant improvement of this result by considering the average of p′p'-degrees of irreducible characters. We also consider fields of character values and prove several improvements of earlier related results.Comment: 23 page

    Low-dimensional complex characters of the symplectic and orthogonal groups

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    We classify the irreducible complex characters of the symplectic groups Sp2n(q)Sp_{2n}(q) and the orthogonal groups Spin2n±(q)Spin_{2n}^\pm(q), Spin2n+1(q)Spin_{2n+1}(q) of degrees up to the bound D, where D=(qn−1)q4n−10/2D=(q^n-1)q^{4n-10}/2 for symplectic groups, D=q4n−8D=q^{4n-8} for orthogonal groups in odd dimension, and D=q4n−10D=q^{4n-10} for orthogonal groups in even dimension.Comment: 44 pages. Comm. Algebra, to appea

    Irreducible restrictions of Brauer characters of the Chevalley group G_2(q) to its proper subgroups

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    Let G2(q)G_2(q) be the Chevalley group of type G2G_2 defined over a finite field with q=p^n elements, where p is a prime number and nn is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of GG in characteristic other than p to a maximal subgroup of G2(q)G_2(q) is still irreducible. Similar results are obtained for 2B2(q)^2B_2(q) and 2G2(q)^2G_2(q).Comment: 30 page

    Character degree sums of finite groups

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    We present some results on character degree sums in connection with certain characteristics of finite groups such as p-solvability, solvability, supersolvability, and nilpotency. Some of them strengthen known results in the literature.Comment: 11 page

    Variations of Landau's theorem for p-regular and p-singular conjugacy classes

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    The well-known Landau's theorem states that, for any positive integer kk, there are finitely many isomorphism classes of finite groups with exactly kk (conjugacy) classes. We study variations of this theorem for pp-regular classes as well as pp-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of pp-regular classes or the number of pp-singular classes of the group. In particular, if GG is a finite group with Op(G)=1O_p(G)=1 then ∣G/F(G)∣p′|G/F(G)|_{p'} is bounded in terms of the number of pp-regular classes of GG. However, it is not possible to prove that there are finitely many groups with no nontrivial normal pp-subgroup and kk pp-regular classes without solving some extremely difficult number-theoretic problems (for instance, we would need to show that the number of Fermat primes is finite).Comment: 23 pages, to appear in Israel J. Mat

    On the average character degree of finite groups

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    We prove that if the average of the degrees of the irreducible characters of a finite group GG is less than 16/5, then GG is solvable. This solves a conjecture of I.M. Isaacs, M. Loukaki, and the first author. We discuss related questions.Comment: The first version is revised based on the referee's report. To appear in Bull. Lond. Math. So

    Abelian subgroups, nilpotent subgroups, and the largest character degree of a finite group

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    Let HH be an abelian subgroup of a finite group GG and π\pi the set of prime divisors of ∣H∣|H|. We prove that ∣HOπ(G)/Oπ(G)∣|H O_{\pi}(G)/ O_{\pi}(G)| is bounded above by the largest character degree of GG. A similar result is obtained when HH is nilpotent.Comment: 16 page

    On the number of conjugacy classes of π\pi-elements in finite groups

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    Let GG be a finite group and π\pi be a set of primes. We show that if the number of conjugacy classes of π\pi-elements in GG is larger than 5/85/8 times the π\pi-part of ∣G∣|G| then GG possesses an abelian Hall π\pi-subgroup which meets every conjugacy class of π\pi-elements in GG. This extends and generalizes a result of W. H. Gustafson.Comment: 7 page

    On the permutation modules for orthogonal groups Om±(3)O_{m}^{\pm}(3) acting on nonsingular points of their standard modules

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    We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups Om±(3)O_{m}^{\pm}(3) with m≥6m\geq6 on nonsingular points of their standard modules. These actions together with those studied in \cite{HN} are all examples of primitive rank 3 actions of finite classical groups on nonsingular points.Comment: 19 pages, 3 table

    Irreducible characters of even degree and normal Sylow 22-subgroups

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    The classical It\^o-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group GG is coprime to a given prime pp, then GG has a normal Sylow pp-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of GG is less than 4/34/3 then GG has a normal Sylow 22-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the It\^o-Michler theorem.Comment: 16 pages. arXiv admin note: text overlap with arXiv:1506.0645
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