Finite groups containing many involutions. (English) Zbl 0811.20024
The author proves the following result: If \(G\) is a finite group and if \(G\) contains at least \(r | G|\) involutions for some real number \(r\), then \(G\) has a normal subgroup \(H\) such that both \(| G:H|\) and \(| H'|\) are bounded by some function of \(r\). The proof of this result relies on known elementary inequalities involving character degrees sums and on a result of P. M. Neumann [Bull. Lond. Math. Soc. 21, 456-458 (1989; Zbl 0695.20018)] which shows that if the number of conjugacy classes of \(G\) is at least \(r | G|\), then the group \(G\) is bounded by abelian by bounded. It is also observed that if the ratio \(T(G)/G\) is bounded below, where \(T(G)\) is the sum of the degrees of the irreducible complex characters of \(G\), then \(G\) is bounded by abelian by bounded and a short proof is given of the converse.
Reviewer: M.Deaconescu (Safat)
MSC:
| 20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
| 20B05 | General theory for finite permutation groups |
| 11N45 | Asymptotic results on counting functions for algebraic and topological structures |
Keywords:
finite group; involutions; character degrees sums; number of conjugacy classes; irreducible complex characters; bounded by abelian by boundedCitations:
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