On a congruence of Blichfeldt concerning the order of finite groups
HTML articles powered by AMS MathViewer
- by David Chillag PDF
- Proc. Amer. Math. Soc. 136 (2008), 1961-1966 Request permission
Abstract:
We show that if $G$ is a finite group, $C$ a conjugacy class of $G$ and $d=\left \vert C\right \vert$, $d_{2},d_{3},\ldots ,d_{m}$ are the distinct elements in the multiset $\left \{ \frac {\left \vert C\right \vert \chi (C)} {\chi (1)}\ |\ \chi \in \mathrm {Irr}(G)\right \}$ (here $\chi (C)$ is the value of $\chi$ on any element of $C$), then \[ \left \vert G/\left \langle C\right \rangle \right \vert \cdot \left ( d-d_{2}\right ) \left ( d-d_{3}\right ) \cdots \left ( d-d_{m}\right ) \equiv 0\ \operatorname {mod}\ \left \vert G\right \vert . \] This is a dual to a generalization of a theorem of Blichfeldt stating that if $G$ is a finite group, $\theta$ a generalized character and $d=\theta (1),d_{2},d_{3},\ldots ,d_{m}$ are the distinct values of $\theta$, then \[ \left \vert \ker (\theta )\right \vert \left ( d-d_{2}\right ) \left ( d-d_{3}\right ) \cdots \left ( d-d_{m}\right ) \equiv 0\ \operatorname {mod} \ \left \vert G\right \vert . \] We also observe that $d=\theta (1)$ in Blichfeldt’s congruence can be replaced, with a minor adjustment, by any rational value of $\theta$. A similar change can be done to the first congruence above.References
- Z. Arad, J. Stavi, and M. Herzog, Powers and products of conjugacy classes in groups, Products of conjugacy classes in groups, Lecture Notes in Math., vol. 1112, Springer, Berlin, 1985, pp. 6–51. MR 783068, DOI 10.1007/BFb0072286
- H. F. Blichfeldt, A theorem concerning the invariants of linear homogeneous groups, with some applications to substitution-groups, Trans. Amer. Math. Soc. 5 (1904), no. 4, 461–466. MR 1500684, DOI 10.1090/S0002-9947-1904-1500684-5
- Peter J. Cameron and Masao Kiyota, Sharp characters of finite groups, J. Algebra 115 (1988), no. 1, 125–143. MR 937604, DOI 10.1016/0021-8693(88)90285-2
- David Chillag, Character values of finite groups as eigenvalues of nonnegative integer matrices, Proc. Amer. Math. Soc. 97 (1986), no. 3, 565–567. MR 840647, DOI 10.1090/S0002-9939-1986-0840647-4
- David Chillag, Regular representations of semisimple algebras, separable field extensions, group characters, generalized circulants, and generalized cyclic codes, Linear Algebra Appl. 218 (1995), 147–183. MR 1324056, DOI 10.1016/0024-3795(93)00167-X
- I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
- Masao Kiyota, An inequality for finite permutation groups, J. Combin. Theory Ser. A 27 (1979), no. 1, 119. MR 541348, DOI 10.1016/0097-3165(79)90012-8
Additional Information
- David Chillag
- Affiliation: Department of Mathematics, Technion, Israel Institute of Technology, Haifa, Israel
- Email: chillag@techunix.technion.ac.il
- Received by editor(s): April 17, 2007
- Published electronically: February 14, 2008
- Communicated by: Jonathan I. Hall
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 1961-1966
- MSC (2000): Primary 20G15
- DOI: https://doi.org/10.1090/S0002-9939-08-09380-5
- MathSciNet review: 2383502