On a congruence of Blichfeldt concerning the order of finite groups

Chillag, David

doi:10.1090/S0002-9939-08-09380-5

On a congruence of Blichfeldt concerning the order of finite groups
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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.

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