A problem in the Theory of Groups and a question related to Fibonacci-Like sequences

This thesis is composed of four chapters, two of which, Chapter 3 and Chapter 4, contain original results. In Chapter 1 we recall some basic notions and establish some of the notation and terminology which will be used in the sequel. For example, we recall some useful results about the class X of groups which are isomorphic to their non-abelian subgroups. Every group of this class is in_nite and 2-generated. This class of groups has been studied by H. Smith and J. Wiegold( [34]). They proved that every insoluble X-group is centre- by-simple and they gave a complete characterization of soluble X-groups. Then we recall some results about _nitely generated groups which are isomorphic to their non-trivial normal subgroups. In particular, we will use the result proved by J.C. Lennox, H. Smith and J. Wiegold in [17], for which if G is a _nitely generated in_nite group that is isomorphic to all its non-trivial normal subgroups and which contains a proper normal subgroup of _nite index, then G ' Z... [edited by author]