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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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