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 thenon-isomorphic irreducible complex representations of g. we resolve for several classes of groups (thecamina p-groups, the generalized camina p-groups, the groups which admit (g;z(g)) as a generalizedcamina pair), the problem of existence of a polynomial f(x) 2 q[x] such that f() = g for someirreducible character  of g. as a consequence, we completely determine the p-groups of order at mostp5 (with p odd) which admit such a polynomial. we deduce the characterization that these are thegroups g for which z(g) is cyclic and (g;z(g)) is a generalized camina pair and, we conjecture thatthis holds good for p-groups of any order.