J. Aust. Math. Soc.  76 (2004), 23-37
On local embedding properties of injectors of finite soluble groups

Stephanie Reifferscheid
  Wilhelm-Schickard Institut für Informatik
  Universität Tübingen
  Sand 14, 72076 Tübingen

In the present paper we consider Fitting classes of finite soluble groups which locally satisfy additional conditions related to the behaviour of their injectors. More precisely, we study Fitting classes $\mathfrak{X}$, $\mathfrak{F}$, $1\not=\mathfrak{X}\subseteq\mathfrak{F}$, such that an $\mathfrak{X}$-injector of $G$ is, respectively, a normal, (sub)modular, normally embedded, system permutable subgroup of $G$ for all $G\in \mathfrak{F}$. Locally normal Fitting classes were studied before by various authors. Here we prove that some important results---already known for normality---are valid for all of the above mentioned embedding properties. For instance, all these embedding properties behave nicely with respect to the Lockett section. Further, for all of these properties the class of all finite soluble groups $G$ such that an $\mathfrak{X}$-injector of $G$ has the corresponding embedding property is not closed under forming normal products, and thus can fail to be a Fitting class.
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