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 ,
,
, such that an
injector of
is, respectively, a normal, (sub)modular,
normally embedded, system permutable subgroup of
for all . Locally normal Fitting classes were studied
before by various authors. Here we prove that
some important resultsalready known for
normalityare 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 such that
an injector of
has the corresponding embedding property is not
closed under forming normal products, and thus
can fail to be a Fitting class.
