ON THE TOTAL CHARACTER OF FINITE GROUPS

For a finite group G, we study the total character G afforded by the direct sum of all the non-isomorphic irreducible complex representations of G. We resolve for several classes of groups (the Camina p-groups, the generalized Camina p-groups, the groups which admit (G;Z(G)) as a generalized Camina pair), the problem of existence of a polynomial f(x) 2 Q[x] such that f() = G for some irreducible character  of G. As a consequence, we completely determine the p-groups of order at most p5 (with p odd) which admit such a polynomial. We deduce the characterization that these are the groups G for which Z(G) is cyclic and (G;Z(G)) is a generalized Camina pair and, we conjecture that this holds good for p-groups of any order.