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

A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group $G$ is 1 or divisible by a prime $p$, then $G$ has a normal $p$-complement. We obtain a significant improvement of this result by considering the average of $p'$-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

We classify the irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and the orthogonal groups $Spin_{2n}^\pm(q)$, $Spin_{2n+1}(q)$ of degrees up to the bound D, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups, $D=q^{4n-8}$ for orthogonal groups in odd dimension, and $D=q^{4n-10}$ 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

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

Character degree sums of finite groups

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

The well-known Landau's theorem states that, for any positive integer $k$, there are finitely many isomorphism classes of finite groups with exactly $k$ (conjugacy) classes. We study variations of this theorem for $p$-regular classes as well as $p$-singular classes. We prove several results showing that the structure of a finite group is strongly restricted by the number of $p$-regular classes or the number of $p$-singular classes of the group. In particular, if $G$ is a finite group with $O_p(G)=1$ then $|G/F(G)|_{p'}$ is bounded in terms of the number of $p$-regular classes of $G$. However, it is not possible to prove that there are finitely many groups with no nontrivial normal $p$-subgroup and $k$ $p$-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

We prove that if the average of the degrees of the irreducible characters of a finite group $G$ is less than 16/5, then $G$ 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

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

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

Let $G$ be a finite group and $\pi$ be a set of primes. We show that if the number of conjugacy classes of $\pi$-elements in $G$ is larger than $5/8$ times the $\pi$-part of $|G|$ then $G$ possesses an abelian Hall $\pi$-subgroup which meets every conjugacy class of $\pi$-elements in $G$. 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

We describe the structure, including composition factors and submodule lattices, of cross-characteristic permutation modules for the natural actions of the orthogonal groups $O_{m}^{\pm}(3)$ with $m\geq6$ 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

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 $G$ is coprime to a given prime $p$, then $G$ has a normal Sylow $p$-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 $G$ is less than $4/3$ then $G$ has a normal Sylow $2$-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