arXiv subjects
Publications and source records attributed to Guohua Qian.
For an irreducible complex character $χ$ of a finite group $G$, the codegree of $χ$ is defined by $|G:\ker(χ)|/χ(1)$, where $\ker(χ)$ is the kernel of $χ$. Given a prime $p$, we provide a classification of finite groups in which every irreducible complex character has either $p'$-degree or $p'$-codegree.
For an irreducible character $χ$ of a finite group $G$, let $\mathrm{cod}(χ):=|G: \ker(χ)|/χ(1)$ denote the codegree of $χ$, and let $\mathrm{cod}(G)$ be the set of irreducible character codegrees of $G$. In this note, we prove that if $\ker(χ)$ is not nilpotent, then there exists an irreducible character $ξ$ of $G$ such that $\ker(ξ)<\ker(χ)$ and $\mathrm{cod}(ξ)> \mathrm{cod}(χ)$. This provides a character codegree analogue of a classical theorem of Broline and Garrison. As a consequence, we obtain that for a nonidentity solvable group $G$, its Fitting height $\ell_{\mathbf{F}}(G)$ does not exceed $|\mathrm{cod}(G)|-1$. Additionally, we provide two other upper bounds for the Fitting height of a solvable group $G$ as follows: $\ell_{\mathbf{F}}(G)\leq \frac{1}{2}(|\mathrm{cod}(G)|+2)$, and $\ell_{\mathbf{F}}(G)\leq 8\log_2(|\mathrm{cod}(G)|)+80$.
In this paper, we get the sharp bound for $|G/O_p(G)|_p$ under the assumption that either $p^2 \nmid χ(1)$ for all $χ\in {\rm Irr}(G)$ or $p^2 \nmid ϕ(1)$ for all $ϕ\in {\rm IBr}_p(G)$. This would settle two conjectures raised by Lewis, Navarro, Tiep, and Tong-Viet.
Let $p$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite nonabelian group $G$. Let $bcl(G)$ be the size of the largest conjugacy class of the group $G$. We show that $|P/O_p(G)| < bcl(G)$ if $G$ is not abelian.
Let $G$ be a finite group and $Irr(G)$ the set of irreducible complex characters of $G$. Let $e_p(G)$ be the largest integer such that $p^{e_p(G)}$ divides $χ(1)$ for some $χ\in Irr(G)$. We show that $|G:\mathbf{F}(G)|_p \leq p^{k e_p(G)}$ for a constant $k$. This settles a conjecture of A. Moretó. We also study the related problems of the $p$-parts of conjugacy class sizes of finite groups.
In this paper we classify the finite groups satisfying the following property $P_4$: their orders of representatives are set-wise relatively prime for any 4 distinct non-central conjugacy classes.
The aim of this paper is to classify the finite nonsolvable groups in which every irreducible character of even degree vanishes on at most two conjugacy classes. As a corollary, it is shown that $L_2(2^f)$ are the only nonsolvable groups in which every irreducible character of even degree vanishes on just one conjugacy class.
In this paper, we prove that if $G$ is a finite group in which elements of the same order outside the center are conjugate, then $G$ is abelian or is isomorphic to $S_3$, the symmetry group of degree 3.