On quasinormal Fitting classes

Abstract

Let Zp be a group of order p and G≀Zp the regular wreath product of the group G with Zp. A Fitting class F is said to be X -quasinormal if there exists a natural number m such that Gm≀Zp∈F whenever F⊆X , p is a prime, G∈F and G≀Zp∈X . In this paper, we generalize the well known theorem of Blessenohl and Gaschütz and prove that the intersection of any set of nonidentity X -quasinormal Fitting classes is a nonidentity X -quasinormal Fitting class. For a Fitting class F , we also discuss the relation between its X -normality and its X -quasinormality, and give the condition of X∗=X⊛ , where “∗ ” is Lockett star operation and X⊛ is the smallest nonidentity X -quasinormal Fitting class.