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On the total character of finite groups | ||
| International Journal of Group Theory | ||
| مقاله 5، دوره 3، شماره 3، آذر 2014، صفحه 47-67 اصل مقاله (389.51 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2014.4446 | ||
| نویسندگان | ||
| Sunil Kumar Prajapati* 1؛ Balasubramanian Sury2 | ||
| 1NBHM Postdoctoral fellow in Indian Statistical Institute Bangalore (I have submitted my PhD thesis at Indian Institute of Technology Delhi). | ||
| 2Indian Statistical Institute bangalore, India | ||
| چکیده | ||
| 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. | ||
| کلیدواژهها | ||
| Finite groups؛ Group Characters؛ Total Characters | ||
| مراجع | ||
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