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 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) \in \mathbb{Q}[x] such that f(\chi) = \tau_G for‎ ‎some irreducible character \chi of G‎. ‎As a consequence‎, ‎we‎ ‎completely determine the p-groups of order at most p^۵ (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.

Keywords:

Finite groups , Group Characters , Total Characters