On the total character of finite groups

Document Type : Research Paper


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


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.


Main Subjects

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