Given a finitely presented group (fp-group) about which nothing is known, the immediate problems are to determine whether it is trivial, finite, infinite, free etc. and to determine its finite homomorphic images, finite index subgroups and so on. The central strategy for analyzing an fp-group is to attempt to construct non-trivial homomorphisms, which may be onto an abelian group, p-group, nilpotent group, soluble group, permutation group (the Todd-Coxeter algorithm) or matrix group (vector enumeration).