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On the orders of vanishing elements of finite groups. (English) Zbl 07341190
An element $$x$$ of a finite group $$G$$ is called a vanishing element if $$\chi(x) = 0$$ for some irreducible character $$\chi$$ of $$G$$. The paper under review is concerned with $$\mathrm{Vo}(G)$$, the set of orders of the vanishing elements in $$G$$. Let $$p$$ be a fixed prime number.
In Theorem A, the author deals with the case where $$\mathrm{Vo}(G)$$ contains precisely one number not divisible by $$p$$. He proves that $$G$$ is solvable and that – under certain additional hypotheses – the $$p$$-length of $$G$$ is at most 2.
In Theorem B, the author considers the situation where $$G$$ is solvable and $$\mathrm{Vo}(G)$$ contains exactly one number divisible by $$p$$. He proves that the commutator subgroup of a Sylow $$p$$-subgroup $$P$$ of $$G$$ is subnormal in $$G$$ and that $$P/O_p(G)$$ is cyclic.
In Theorem C, the author supposes that $$p>7$$ and that $$\mathrm{gcd}(a,b)$$ is a power of $$p$$ whenever $$a,b \in \mathrm{Vo}(G)$$ are distinct. He shows that $$G$$ is solvable and that the $$p$$-length of $$G$$ is at most 2.
Finally, in Theorem D, the author describes the finite solvable groups $$G$$ with the property that the numbers in $$\mathrm{Vo}(G)$$ are prime powers.
##### MSC:
 20C15 Ordinary representations and characters
##### Keywords:
irreducible character; vanishing element
GAP
Full Text:
##### References:
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