We have asserted, more than once, that absolute irreducibility is a much stronger condition than irreducibility. In this chapter, at long last, we really substantiate this claim. In general terms our contention is that a reasonable absolutely irreducible group should be abelian by locallyfinite at least. Further, the same conclusion should hold for a reasonable normal subgroup H of an absolutely irreducible group G. Moreover H should be more or less rigid, in that G/CG(H) should be much smaller than one might at first expect. What groups are reasonable in this context? Clearly a group with a non-cyclic free subgroup is unreasonable. We are a long way from proving the converse of this, if indeed it is actually true.

The main sections of this chapter are Section 5.4, where we consider locally finite normal subgroups, and Section 5.6, where soluble normal subgroups are considered. We need to start, in Section 5.1, with a close look at the linear case and, since it is required for Section 5.6, in Section 5.5 we study what happens when the ground division ring is locally finite-dimensional over its centre. Sections 5.2 and 5.3 are service sections, of a mainly ring-theoretic nature, for what comes later.