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On the orders of vanishing elements of finite groups. (English) Zbl 07341190
Summary: Let \(G\) be a finite group and \(p\) be a prime. Let \(\operatorname{Vo}(G)\) denote the set of the orders of vanishing elements, \( \operatorname{Vo}_p(G)\) be the subset of \(\operatorname{Vo}(G)\) consisting of those orders of vanishing elements divisible by \(p\) and \(\operatorname{Vo}_{p^\prime}(G)\) be the subset of \(\operatorname{Vo}(G)\) consisting of those orders of vanishing elements not divisible by \(p\). Dolfi, Pacifi, Sanus and Spiga proved that if \(a\) is not a \(p\)-power for all \(a \in \operatorname{Vo}(G)\), then \(G\) has a normal Sylow \(p\)-subgroup. In another article, the same authors also show that if \(\operatorname{Vo}_{p^\prime}(G) = \varnothing \), then \(G\) has a normal nilpotent \(p\)-complement. These results are variations of the well known Ito-Michler and Thompson theorems. In this article we study solvable groups such that \(| \operatorname{Vo}_p(G) | = 1\) and show that \(P^\prime\) is subnormal. This is analogous to the work of Isaacs, Moréto, Navarro and Tiep where they considered groups with just one character degree divisible by \(p\). We also study certain finite groups \(G\) such that \(| \operatorname{Vo}_{p^\prime}(G) | = 1\) and we prove that \(G\) has a normal subgroup \(L\) such that \(G / L\) a normal \(p\)-complement and \(L\) has a normal \(p\)-complement. This is analogous to the recent work of Giannelli, Rizo and Schaeffer Fry on character degrees with a few \(p^\prime \)-character degrees. Bubboloni, Dolfi and Spiga studied finite groups such that every vanishing element is of order \(p^m\) for some integer \(m \geqslant 1\). As a generalization, we investigate groups such that \(\gcd(a, b) = p^m\) for some integer \(m \geqslant 0\), for all \(a, b \in \operatorname{Vo}(G)\). We also study finite solvable groups whose irreducible characters vanish only on elements of prime power order.
20C15 Ordinary representations and characters
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