Groups Whose Nonlinear Irreducible p-Brauer Characters are Real Valued

Let p be a given prime. A finite group G whose all nonlinear irreducible p-Brauer characters are real valued is called a p-regular (1)-group. The aim of this article is to show some results about the structures of p-regular (1)-groups. In particular, we will build the connections between p-regular (1)-groups and p-modular Frobenius groups. If these results apply to a finite group G with p inverted iota|G|, we obtain similar results about finite groups whose nonlinear irreducible characters are real valued, and establish connections between these groups and Frobenius groups.