  The article $-$ part of a larger thesis which aims to give a detailed
description of the generalisation to the category of groups with operators of
the classical theory of semisimplicity for modules $-$ presents a
straightforward generalisation to groups with operators of a number of
invariants well-known in the theory of modules, having a special bearing on
phenomena of semisimplicity. The behaviour of these generalised invariants in
relation to the fundamental construction of restricted direct sums and in
particular in the context of semisimple groups with operators is in particular
examined. The article also introduces a particular type of morphisms between
groups with operators, morphisms which naturally preserve the generalised
invariants in question and thus show themselves to be an adequate notion for
the study of semisimplicity in the more general frame considered.
