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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
Software:
GAP
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