ABSTRACT
In Chapter 6 we discuss various finiteness conditions on the size of the G-orbits of G. The first situation that arises concerns the case when a G = {ga|g ∈ G} is finite for all a ∈ A, corresponding to the situation with FC-groups. Such groups G that have this property for all elements a A are said to have finite orbits and are precisely the residually finite groups. If there is a bound d on the size of a G for all a A, then G has boundedly finite orbits and in this case much more can be said. For example, when F has characteristic 0 and G has boundedly finite orbits, then G must be finite. We then discuss groups with finite dimensional orbits, so now FaG is finite dimensional. In this case we obtain residually (finite dimensional) linear groups and again much more can be said if there is a bound on the dimension of FaG for all a A. In this case, for example, periodic subgroups of GL(F, A) having boundedly finite dimensional orbits are locally finite. Groups with finite orbits of subspaces are also discussed. The chapter ends with a discussion of the sizes of dim F (GB/B) and dim F (B/core G B). When B is G-invariant, these dimensions are both 0. Thus the values of dim F (GB/B) (the upper measure of non-G-invariance) and dim F (B/core G B) (the lower measure of non-G-invariance) give some indication about how close B is to being G-invariant or how far B is from being G-invariant.
