Finite groups with some restriction on the vanishing set. (English) Zbl 1498.20020
Consider a finite group \(G\) such that \(\mathrm{gcd}(\mathrm{ord}(x),\mathrm{ord}(y))\leq n\) for any two vanishing elements \(x\) and \(y\) contained in distinct conjugacy classes. In this paper, the authors show that
- i)
- if \(n=1\), then \(G\) is a Frobenius group with an abelian kernel and complement of order two.
- ii)
- if \(n=2\), then \(G\) is solvable.
- iii)
- if \(n=2\) and \(G\) is supersolvable, then \(G\) has a normal metabelian \(2\)-complement.
Reviewer: Mahmood Robati Sajjad (Qazvin)
MSC:
| 20C15 | Ordinary representations and characters |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
Software:
GAPReferences:
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