The realization of the Total Character (or Gel’fand Character) τG of a finite group G, i.e. the sum of all ordinary irreducible characters of G is an old problem in character theory of finite groups. One possible approach is to try to realize τG as a polynomial in some irreducible character of G. In this vein, K. W. Johnson has asked whether it is possible to express the total character of G as a polynomial, with integer coefficients, in a single irreducible character of G. We study for several classes of finite nilpotent groups, the problem of existence of a polynomial f(x) ∈ Q[x] such that f(χ) = τG for some irreducible character χ of G. As a consequence, we completely determine the p-groups of order at most p5 (with p odd) which admit such a polynomial. Indeed, we prove that: If G is a non-abelian p-group of order p5, then G has such a polynomial if and only if Z(G) is cyclic and (G,Z(G)) is generalized Camina pair and, we conjecture that this holds good for p-groups of any order. In the talk, we also discuss about the nonlinear irreducible characters of p-groups of order at most p5.
School of Mathematical Sciences
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