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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a congruence of Blichfeldt concerning the order of finite groups


Author: David Chillag
Journal: Proc. Amer. Math. Soc. 136 (2008), 1961-1966
MSC (2000): Primary 20G15
Published electronically: February 14, 2008
MathSciNet review: 2383502
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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)} \vert \chi\in \mathrm{Irr}(G)\right\}$ (here $ \chi(C)$ is the value of $ \chi$ on any element of $ C$), then

$\displaystyle \left\vert G/\left\langle C\right\rangle \right\vert \cdot\left( ... ...cdots\left( d-d_{m}\right) \equiv0 \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

$\displaystyle \left\vert \ker(\theta)\right\vert \left( d-d_{2}\right) \left( d... ...dots\left( d-d_{m}\right) \equiv0 \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.


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Additional Information

David Chillag
Affiliation: Department of Mathematics, Technion, Israel Institute of Technology, Haifa, Israel
Email: chillag@techunix.technion.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09380-5
PII: S 0002-9939(08)09380-5
Received by editor(s): April 17, 2007
Published electronically: February 14, 2008
Communicated by: Jonathan I. Hall
Article copyright: © Copyright 2008 American Mathematical Society