Invariants of Algebras and Finite Groups

Group theoretical constructions usually terminate in the problem to decide whether two groups are isomorphic. In the case of arbitrary finite groups the calculation of ordinary group characters is not sufficient to decide about it. R. Brauer posed the problem, to find suitable additional group invariants. Applying the theory of norm-type forms of associative algebras, specialized to group algebras, we found, that the 1-, 2-, and 3-characters in the nonmodular case (with some restrictions on the characteristic), especially over the field of complex numbers, are necessary and sufficient for the finite groups. This sharpens a recent result of E. Formanek and D. Sibley on group determinants. Detailed proofs will be given elsewhere. Here we give an overview on recent related results and add a remark concerning calculations in the modular group case.