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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.
##### MSC:
 20C15 Ordinary representations and characters
GAP
Full Text:
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