@article {
author = {Prajapati, Sunil and Sury, Balasubramanian},
title = {On the total character of finite groups},
journal = {International Journal of Group Theory},
volume = {3},
number = {3},
pages = {47-67},
year = {2014},
publisher = {University of Isfahan},
issn = {2251-7650},
eissn = {2251-7669},
doi = {10.22108/ijgt.2014.4446},
abstract = {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.},
keywords = {Finite groups,Group Characters,Total Characters},
url = {https://ijgt.ui.ac.ir/article_4446.html},
eprint = {https://ijgt.ui.ac.ir/article_4446_e0d321ff268fc949ff98a187267f48e3.pdf}
}