Hereditarily Minimal Topological Groups

A Hausdorff topological group G is minimal(D.Doitchinov[17]and Stephen-son[43])if every injective continuous group homomorphism G→P into a Hausdorff topological group is a topological embedding.In the category of groups,bijective ho-momorphisms is equivalent to isomorphisms.In the category of topological groups,a continuous bijective homomorphism is not necessary to be a topological isomor-phisms,while a minimal topological group satisfies this property.In other words,a continuous bijective homomorphism becomes a topological isomorphisms for a min-imal topological group.Total minimality is stronger than minimality,we call a group G totally minimal if all Hausdorff quotients of G are minimal,is nothing else but the open mapping property([12]).The class of totally minimal groups witnesses the deep roots of this notion in Analysis[22],as well as Algebra[5](via the discrete minimal groups,usually called also non-topologizable groups[25,35,41],and more general-ly,through the algebraic structure of the minimal groups).In 1972,Prodanov proved that an infinite compact abelian group that all subgroups are minimal is precisely the group Zp of p-adic integers,which connected the miniimal groups and Number Theo-ry.In this thesis,we mostly generalize the Prodanov’s Theorem,also pose some open questions which deserve to be done in further research.The contents are as follows:In preface,we introduce the concepts and backgrounds of(locally)minimal groups,the main results,and notations used in this thesis.In chapter 1,we study locally compact groups having all subgroups(locally)minimal,such groups are called hereditarily(locally)minimal.First,we prove a conjecture in[48],showing that the group Qp(?)Qp*is hereditarily locally minimal,where Qp*is the multiplicative group of non-zero p-adic numbers acting on the first component by multiplication.Then,we extend Prodanov’s theorem to the non-abelian case at several levels.For infinite hypercentral(in particular,nilpotent)locally compact groups,we show that the hereditarily minimal ones remain the same as in the abelian case.In other words,an infinite hypercentral locally compact groups is hereditarily minimal if and only if it is isomorphic to Zp for some prime p.In chapter 2,we focus on the hereditary minimality in locally compact solvable case.We begin this chapter with some general results on the semidirect products of p-adic integers.Then we classify completely the hereditarily minimal locally compact solvable groups.As a corollary of it,we also show that such groups are even compact and metabelian.In chapter 3,we study locally compact groups having all dense subgroups(lo-cally)minimal,such groups are called densely(locally)minimal.We prove that in case that a topological abelian group G is either compact or connected locally compact,G is densely locally minimal if and only if G either is a Lie group or has an open subgroup isomorphic to Zp for some prime p.Our Theorem 3.1.8 provides another extension of Prodanov’s theorem:an infi-nite locally compact abelian group is densely minimal if and only if it is isomorphic to Zp.For the non-abelian case,we show that there exists a densely minimal,com-pact,two-step nilpotent group that neither is a Lie group nor has an open subgroup isomorphic to Zp.