Finite groups with a unique nonlinear nonfaithful irreducible character

Ali Iranmanesh and Amin Saeidi

Department of Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box: 14115-137, Tehran, Iran


Abstract: 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.

AMSclassification: primary 20C15; secondary 20D15, 20F16.

Keywords: minimal normal subgroups, faithful characters, strong condition on normal subgroups, Frobenius groups.