ON THE TOTAL CHARACTER OF FINITE GROUPS

For a finite group G, we study the total character tau G afforded by the direct sum of all the non-isomorphic irreducible complex representations of G. We resolve for several classes of groups (the Camilla p-groups, the generalized Camilla p-groups, the groups which admit (G, Z(G)) as a generalized Camilla pair), the problem of existence of a polynomial f (x) is an element of Q[x] such that f (x) = tau G for some irreducible character x of G. As a consequence, we completely determine the p-groups of order at most p(5) (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 Camilla pair and, we conjecture that this holds good for p-groups of any order.