Abstract
An irreducible character χ of a finite group G is called a Heisenberg character if Kerχ ⊇ [G, [G,G]]. In this paper, we prove that the group G has exactly r, r ≤ 3, Heisenberg characters if and only if |G/G′| = r. If G has exactly four Heisenberg characters, then |G/G′| = 4, but the converse is not correct in general. Finally, it is proved that if G has exactly five Heisenberg characters, then |G/G′| = 5 or |G/G′| = 4 and one of the Heisenberg characters of G has the degree 2.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 177, Algebra, 2020.
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Zolfi, A., Ashrafi, A.R. On the Number of Heisenberg Characters of Finite Groups. J Math Sci 275, 674–682 (2023). https://doi.org/10.1007/s10958-023-06708-3
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DOI: https://doi.org/10.1007/s10958-023-06708-3