Quasinormal Fitting classes of finite groups

Let $\mathbb{P}$ be the set of all primes, $Z_{n}$ a cyclic group of order $n$ and $X ~wr ~Z_{n}$ the regular wreath product of the group $X$ with $Z_{n}$. A Fitting class $\mathfrak{F}$ is said to be $\mathfrak{X}$-quasinormal (or quasinormal in a class of groups $\mathfrak{X}$) if $\mathfrak{F}\subseteq \mathfrak{X}$ is a prime, groups $G\in \mathfrak{F}$ and $G ~wr ~Z_{p}\in \mathfrak{X}$, then there exists a natural number $m$ such that $G^{m} ~wr ~Z_{p}\in \mathfrak{F}$. If $\mathfrak{X}$ is the class of all soluble groups, then $\mathfrak{F}$ is normal Fitting class. In this paper we generalize the well-known theorem of Blessenohl and Gaschutz in the theory of normal Fitting classes. It is proved, that the intersection of any set of nontrivial $\mathfrak{X}$-quasinormal Fitting classes is a nontrivial $\mathfrak{X}$-quasinormal Fitting class. In particular, there exists the smallest nontrivial $\mathfrak{X}$-quasinormal Fitting class. We confirm a generalized version of the Lockett conjecture about the structure of a Fitting class for the case of $\mathfrak{X}$-quasinormal classes, where $\mathfrak{X}$ is a local Fitting class of partially soluble groups.