The Residual Properties And Separability Properties Of Groups

In 1911, M. Dehn gave three fundamental problems in combinatorial group theory: word problem, conjugacy problem and isomorphism problem. Since finitely presented, residually finite(conjugacy separable, subgroup separable) groups have solvable word problem(conjugacy problem, generalize word problem, respectively), the residual properties and separability properties are important for combinatorial group theory.The residual properties and separability properties are also interesting properties in abstract group theory. Moreover we can define a profinite topology in the group, then some residual properties and separability properties become the topological properties. We can study these properties in this direction.In chapter 1, we give the definitions and examples of groups with residual properties and separability properties.In chapter 2, we consider some special separability properties. These properties will be needed in the following chapter.In chapter 3, we study the properties of residual finite groups and residual finite p-groups. We get criterion for generalized free products to be residual finite groups and residual finite p-groups. And we get a condition for a polygonal products to be residually finite.In chapter 4, we study the cyclic subgroup separability(Ï€_c). We get criterions for generalize free products and HNN-extension to be cyclic subgroup separable.In chapter 5, the conjugacy separability is considered. And we get a criterion for the cyclic conjugacy separability of generalized free products.In the last chapter, we discuss the affection of the number of non-power subgroups. We prove: In a non-cyclic group, if the number of non-power subgroups is finite, then the group is finite. And following this theorem, we get a classification of groups.