Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that χ(g) ≠ 0 for all irreducible characters χ of G. Such an element is said to be non-vanishing inG. Let p be a prime. If all p-elements of N satisfy the previous property, then we prove that N has a normal Sylow p-subgroup. As a consequence, we also study certain arithmetical properties of the G-conjugacy class sizes of the elements of N which are zeros of some irreducible character of G. In particular, if N= G, then new contributions are obtained.
Felipe, M. J., Grittini, N., Sotomayor, V., On zeros of irreducible characters lying in a normal subgroup, <<ANNALI DI MATEMATICA PURA ED APPLICATA>>, 2020; 199 (5): 1777-1787. [doi:10.1007/s10231-020-00942-1] [http://hdl.handle.net/10807/188780]
On zeros of irreducible characters lying in a normal subgroup
Grittini, Nicola;
2020
Abstract
Let N be a normal subgroup of a finite group G. In this paper, we consider the elements g of N such that χ(g) ≠ 0 for all irreducible characters χ of G. Such an element is said to be non-vanishing inG. Let p be a prime. If all p-elements of N satisfy the previous property, then we prove that N has a normal Sylow p-subgroup. As a consequence, we also study certain arithmetical properties of the G-conjugacy class sizes of the elements of N which are zeros of some irreducible character of G. In particular, if N= G, then new contributions are obtained.
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