%0 Journal Article
%T On the total character of finite groups
%J International Journal of Group Theory
%I University of Isfahan
%Z 2251-7650
%A Prajapati, Sunil Kumar
%A Sury, Balasubramanian
%D 2014
%\ 09/01/2014
%V 3
%N 3
%P 47-67
%! On the total character of finite groups
%K Finite groups
%K Group Characters
%K Total Characters
%R 10.22108/ijgt.2014.4446
%X 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 $p^5$ (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 Camina pair and, we conjecture that this holds good for $p$-groups of any order.
%U https://ijgt.ui.ac.ir/article_4446_e0d321ff268fc949ff98a187267f48e3.pdf