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

Y. Berkovich (2008). Groups of prime power order. {V}ol. 1. of de Gruyter Expositions in Mathematics, With a foreword by Zvonimir Janko, Walter de Gruyter GmbH \& Co. KG, Berlin. 46 A. R. Camina (1978). Some conditions which almost characterize Frobenius group. Israel J. Math.. 31, 153-160 R. Dark and C. M. Scoppola (1996). On Camina groups of prime power order. J. Algebra. 181, 787-802 S. M. Gagola, Jr. and M. L. Lewis (1998). Squares of characters that are the sum of all irreducible characters. Illinois J. Math.. 42 (4), 655-672 R. Gow (1983). Properties of the finite linear group related to the transpose-inverse involution. Proc. London Math. Soc.. 47 (3), 493-506 P. Hall (1940). The classification of prime-power groups. J. Reine Angew. Math.. 182, 130-141 I. M. Isaacs (2000). Character theory of finite groups. MS Chelsea Publishing, Academic Press, New York. R. James (1980). The groups of order p^6 (p an odd prime. Math. Comp.. 34 (150), 613-637 G. Karpilovsky (1992). Group representations. {V}ol. 1. {P}art {B}. North-Holland Mathematics Studies, of North-Holland Publishing Co., Amsterdam, Introduction to group representations and characters, i-xiv. 175, 621-1274 V. Kodiyalam and D. N. Verma A natural representation model for symmetric groups. http://arxiv.org/pdf/math/0402216v1.pdf. M. L. Lewis, A. Moreto and T. R. Wolf (2005). Non-divisibility among character degrees. J. Group Theory. 8, 561-588 M. L. Lewis (2009). The vanishing-off subgroup. J. Algebra. 321 (4), 1313-1325 M. L. Lewis (2009). Generalizing Camina groups and their character tables. J. Group Theory. 12, 209-218 M. L. Lewis (2012). On p-group Camina pairs. J. Group Theory. 15, 469-483 I. D. Macdonald (1981). ome p-groups of Frobenius and extra-special type. Israel J. Math.. 40, 350-364 I. D. Macdonald (1986). More on p-groups of Frobenius type. Israel J. Math.. 56, 335-344 S. Mattarei (1992). Character tables and Metabelian groups. J. London Math. Soc.. 46 (2), 92-100 E. Poimenidou and H. Wolfe (2003). Total characters and {C}hebyshev polynomials. Int. J. Math. Math. Sci.. 38, 2447-2453 S. K. Prajapati and R. Sarma On the Gel'fand Character. submitted for publication.