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minimal normal subgroups; faithful characters; strong condition on normal subgroups; Frobenius groups
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.
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