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Average number of zeros of characters of finite groups. (English) Zbl 07335749
Given a finite group \(G\), let \(\mathrm{anz}(G)\) be the average number of zeros in the rows of the character table. The main result in this paper shows that if \(\mathrm{anz}(G)<1\) then \(G\) is solvable. In order to prove this result, the author proves the following extendibility result: if \(N\) is a nonabelian minimal normal subgroup of a finite group \(G\), then there exists a nonprincipal irreducible character of \(N\) that extends to an irreducible character \(\chi\) of \(G\) that vanishes on at least two \(G\)-conjugacy classes. As could be expected, the proof of this result relies on the classification of finite simple groups.
Other related results are proved in this paper. For instance, if \(\mathrm{anz}(G)<1/2\) then \(G\) is supersolvable and if \(G\) has odd order and \(\mathrm{anz}(G)<1\) then \(G\) is supersolvable. It is also conjectured that this later result can be improved to: if \(G\) has odd order and \(\mathrm{anz}(G)<16/11\) then \(G\) is supersolvable.
Reviewer’s remarks: The reviewer has proved that there are exactly 4 nonabelian odd order groups with \(\mathrm{anz}(G)<16/11\): they are the Frobenius groups of order \(3\cdot7\), \(3\cdot13\), \(3\cdot19\) and \(5\cdot11\). He has also obtained a classification of the finite groups with \(\mathrm{anz}(G)<1\) that is independent of the main result of this paper and does not use the above mentioned extendibility theorem. [A. Moretó, “Groups with a small average number of zeros in the character table”, Preprint, arXiv:2106.01943].
MSC:
20C15 Ordinary representations and characters
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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