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In this note, we study finite groups possessing exactly one nonlinear non-faithful irreducible character. Our main result is to classify solvable groups that satisfy this property. Also, we give examples to show that these groups need not to be solvable in general.

Cliques of irreducible representations of p-solvable groups and a theorem analogous to Alperin's conjecture.

Harald Ellers (1994)

Mathematische Zeitschrift

Closure Properties for Fitting Functors.

James C. Beidleman, Peter Hauck (1989)

Monatshefte für Mathematik

CLT and non-CLT groups of order $p^{2} q^{2}$

S. Baskaran (1976)

Fundamenta Mathematicae

Cohomological characterisations of saturated formations and homomorphs of finite groups.

U. Stammbach, D.W. Barnes, P. Schmid (1978)

Commentarii mathematici Helvetici

Complete metacyclic groups

N. J. Mutio (1975)

Commentationes Mathematicae Universitatis Carolinae

Conditions for p-supersolubility and p-nilpotency of finite soluble groups

Wenai Yan, Baojun Li, Zhirang Zhang (2013)

Colloquium Mathematicae

Let ℨ be a complete set of Sylow subgroups of a group G. A subgroup H of G is called ℨ-permutably embedded in G if every Sylow subgroup of H is also a Sylow subgroup of some ℨ-permutable subgroup of G. By using this concept, we obtain some new criteria of p-supersolubility and p-nilpotency of a finite group.

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Blocks with Dihedral Defect Groups in Solvable Groups.

$c$ -semipermutable subgroups of finite groups.

Caractères venant des ...-normalisateurs d'une groupe fini résoluble.

Caratterizzazione dei gruppi risolubili d'ordine finito complementati

Catene pronormali nei gruppi finiti supersolubili

Characterization of $r$ -solvable groups.

Characterization of the soluble one-generator totally saturated formations of finite groups.

Characters of Groups with Normal Extra Special Subgroups.

Charakterisierungen einiger Schunckklassen endlicher auflösbarer Gruppen II.

Classification of finite groups satisfying a minimal condition.

Classification of solvable groups possessing a unique nonlinear non-faithful irreducible character