University of IsfahanInternational Journal of Group Theory2251-76503320140901On the total character of finite groups4767444610.22108/ijgt.2014.4446ENSunil KumarPrajapatiNBHM Postdoctoral fellow in Indian Statistical Institute Bangalore (I have submitted my PhD thesis at Indian Institute of Technology Delhi).BalasubramanianSuryIndian Statistical Institute bangalore, IndiaJournal Article20130722For 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.https://ijgt.ui.ac.ir/article_4446_e0d321ff268fc949ff98a187267f48e3.pdf