Two Properties of Brauer Character Tables

The aim of this note is to characterize some quantitative properties in a finite group by using modular representation theory.Utilizing the Schur-Zassenhaus theorem and modular representation theory,the authors prove in this note that any prime p does not divide the number of p-regular elements of a finite group.According to the Brauer character table of a finite group,some information on number theory can be obtained.Using modular representation theory and Galois theory,the authors present in this note that every row sum in the Brauer character table of any finite group is a rational integer.In addition,if a finite group is p-solvable,then each column sum in its Brauer character table is a rational integer.