Geodesic
Finite groups with a unique nonlinear nonfaithful irreducible character
Archivum mathematicum, Tome 47 (2011) no. 2, pp. 91-98.

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In this paper, we consider finite groups with precisely one nonlinear nonfaithful irreducible character. We show that the groups of order 16 with nilpotency class 3 are the only $p$-groups with this property. Moreover we completely characterize the nilpotent groups with this property. Also we show that if $G$ is a group with a nontrivial center which possesses precisely one nonlinear nonfaithful irreducible character then $G$ is solvable.
Classification : 20C15, 20D15, 20F16
Keywords: minimal normal subgroups; faithful characters; strong condition on normal subgroups; Frobenius groups 
@article{ARM_2011__47_2_a2,
     author = {Iranmanesh, Ali and Saeidi, Amin},
     title = {Finite groups with a unique nonlinear nonfaithful irreducible character},
     journal = {Archivum mathematicum},
     pages = {91--98},
     publisher = {mathdoc},
     volume = {47},
     number = {2},
     year = {2011},
     mrnumber = {2813535},
     zbl = {1249.20009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2011__47_2_a2/}
}
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Iranmanesh, Ali; Saeidi, Amin. Finite groups with a unique nonlinear nonfaithful irreducible character. Archivum mathematicum, Tome 47 (2011) no. 2, pp. 91-98. http://geodesic.mathdoc.fr/item/ARM_2011__47_2_a2/