ﻻ يوجد ملخص باللغة العربية
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 mainly discuss the characterization of a class of arithmetic functions $f: N rightarrow C$ such that $f(u^{2}+kv^2)=f^{2}(u)+kf^{2}(v)$ $(k, u, v in N)$. We obtain a characterization with given condition, propose a conjecture and show the result holds for $k in {2, 3, 4, 5 }$.
Residual finiteness growth measures how well-approximated a group is by its finite quotients. We prove that some related growth functions characterize linearity for a class of groups including all hyperbolic groups.
In this paper, we get the sharp bound for $|G/O_p(G)|_p$ under the assumption that either $p^2 mid chi(1)$ for all $chi in {rm Irr}(G)$ or $p^2 mid phi(1)$ for all $phi in {rm IBr}_p(G)$. This would settle two conjectures raised by Lewis, Navarro, Tiep, and Tong-Viet.
Using computational methods, we complete the determination of the $3$-modular character table of the Chevalley group $F_4(2)$ and its covering group.
Generalizing Block and Weinbergers characterization of amenability we introduce the notion of uniformly finite homology for a group action on a compact space and use it to give a homological characterization of topological amenability for actions. By