Two groups with isomorphic group algebras
Suppose that G and H are finite groups. If \({\mathbb{C}}G\cong {\mathbb{C}}H\), I. M. Isaacs proved that the nilpotency of G implies the nilpotency of H. In this paper it is proved the modular version of the above result, when \({\mathbb{C}}\) is replaced by an algebraically closed field of positive characteristic. Also, a well known Theorem of Thompson on character degrees is obtained for Brauer characters: Let G be a p-solvable group and \(q\neq p\) a prime number. If q divides \(\phi\) (1) for all \(\phi\) nonlinear irreducible p-Brauer characters of G, then G has a normal q- complement.
scientific article; zbMATH DE number 4103324