On groups all of whose irreducible characters are quasiprimitive. (Über Gruppen, deren irreduzible Charaktere sämtlich quasiprimitiv sind.) (German) Zbl 0874.20005

Bartsch, R.; Pazderski, G.

Sitz.ber. Math.-Nat.wiss. Kl., Akad. Gem.nütz. Wiss. Erfurt 7, 1-17 (1996).

A finite group \(G\) is called quasi-primitive (characteristically quasi-primitive) if, for every irreducible character \(\chi\) of \(G\) and every normal (characteristic) subgroup \(N\) of \(G\), the restriction \(\chi_N\) is a multiple of an irreducible character of \(N\). The authors show that a finite group \(G\) is quasi-primitive if and only if \(G/Z(G)\) is a direct product of nonabelian finite simple groups. Their proof involves the classification of finite simple groups. They also show that a characteristically quasi-primitive finite solvable group \(G\) is nilpotent of class at most 2, with \(G'\) a direct product of elementary abelian groups of prime power order.

Reviewer: B.Külshammer (Jena)

MSC:

20C15

Ordinary representations and characters

Keywords:

quasi-primitive characters; finite groups; irreducible characters; characteristically quasi-primitive finite solvable groups