Prajapati, S., Sury, B. (2014). On the total character of finite groups. International Journal of Group Theory, 3(3), 47-67. doi: 10.22108/ijgt.2014.4446

Sunil Kumar Prajapati; Balasubramanian Sury. "On the total character of finite groups". International Journal of Group Theory, 3, 3, 2014, 47-67. doi: 10.22108/ijgt.2014.4446

Prajapati, S., Sury, B. (2014). 'On the total character of finite groups', International Journal of Group Theory, 3(3), pp. 47-67. doi: 10.22108/ijgt.2014.4446

Prajapati, S., Sury, B. On the total character of finite groups. International Journal of Group Theory, 2014; 3(3): 47-67. doi: 10.22108/ijgt.2014.4446

^{1}NBHM Postdoctoral fellow in Indian Statistical Institute Bangalore (I have submitted my PhD thesis at Indian Institute of Technology Delhi).

^{2}Indian Statistical Institute bangalore, India

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.

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