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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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