Abstract
Given that the set of endomorphisms of a group is contained in the set of distributive elements of its endomorphism near-ring, which, in turn, is contained in the endomorphism near-ring, we show that the class of all groups is partitioned into four nonempty subclasses when all combinations of these inclusions, proper or non-proper, are considered. Furthermore, a characterization of each subclass is given in terms of the orbits of the underlying group.
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