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Zeros of characters of finite groups. (English) Zbl 0965.20003
W. Burnside proved that each nonlinear irreducible character of a finite group $$G$$ has a zero. The main result of this note (Theorem B) asserts that every nonlinear irreducible character of $$G$$ has a zero $$g$$ such that $$o(g)$$, the order of $$g$$, is a prime power. It follows from this result that if, in addition, $$\chi(1)$$ is a $$\pi$$-number, then some $$\pi$$-element of $$G$$ is a zero of $$\chi$$ (Theorem A). In the case when $$|\pi|=1$$, the authors present an elementary proof of Theorem A (Theorem 2.3). In the general result, they use the classification of finite simple groups. In particular, the following two deep results, depending on the classification: (i) If $$H<G$$, then the set $$G-(\bigcup_{x\in G}H^x)$$ possesses an element of prime power order (W. M. Kantor and M. Schacher); (ii) Assume that $$G$$ is a quasisimple group and let $$z\in Z(G)$$ be not a commutator. Then $$o(z)\in\{2,4,6,12\}$$ and in all these cases the (simple) groups $$G/Z(G)$$ are listed (H. Blau). A comparatively easy argument reduces Theorem B to the case when $$G$$ is simple.
The following assertion is proved by inspection of the Atlas of finite groups: Theorem 3.4. Let $$G$$ be a sporadic simple group. Then there exist four conjugacy classes of prime power elements in $$G$$ such that every non-trivial irreducible character of $$G$$ vanishes on at least one of them. It follows that Theorem B is true for sporadic simple groups.

##### MSC:
 20C15 Ordinary representations and characters 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D05 Finite simple groups and their classification
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