Character Theory and Group Rings

While we were graduate students, Marty Isaacs and I worked together on the character theory of finite groups, studying in particular the character degrees of finite p-groups. Somewhat later, my interests turned to ring theory and infinite group theory. On the other hand, Marty continued with character theory and soon became a leader in the field. Indeed, he has had a superb career as a researcher, teacher and expositor. In celebration of this, it is my pleasure to discuss three open problems that connect character theory to the ring-theoretic structure of group rings. The problems are fairly old and may now be solvable given the present state of the subject. A general reference for character theory is of course Marty’s book [6], while [10] affords a general reference for group rings. 1. Character Regular p-Groups As is well known, the degrees of the irreducible complex characters of a finite p-group G are all powers of p, and we write e(G) = e if the largest such character degree is equal to p. It is presumably a hopeless task to try to characterize the p-groups G with e(G) equal to a specific number e, but it is possible that certain of these groups, the ones that do not have a maximal subgroup M with e(M) = e− 1, do in fact exhibit some interesting structure. One possible tool to study this situation is based on the following Definition 1.1. If e(G) = e, then G is said to be character regular precisely when G is faithfully embedded in the totality of its irreducible representations of degree p, or equivalently when ⋂