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Unformatted text preview: GLASNIK MATEMATI CKI Vol. 42(62)(2007), 357 362 ON A CHARACTERIZATION OF QUASICYCLIC GROUPS Dabin Zheng, Yujie Ma and Heguo Liu Hu Bei University, Chinese Academy of Sciences, China Abstract. Let G be an infinite solvable group (resp. an infinite group properly containing its commutator subgroup G ). We prove that G is isomorphic to a quasicyclic group if and only if all proper normal subgroups of G are finitely generated (resp. all proper normal subgroups of G are cyclic or finite). In this paper, the symbols Q , Z , N denote the rational numbers, the integers, the nonnegative integers, respectively. A quasicyclic group (or Prufer group) is the pprimary component of Q / Z , that is, the unique maximal psubgroup of Q / Z , for some prime number p . Any group isomorphic to it will also be called a quasicyclic group and denoted by Z p . Quasicyclic groups play an important roles in the infinite abelian group theory. They may also be defined in a number of equivalent ways (again, up to isomorphism): A quasicyclic group is the group of all p nth complex roots of 1, for all n N . A quasicyclic group is the injective hull of Z /p Z (viewing abelian groups as Zmodules). A quasicyclic group is the direct limit of the groups Z /p n Z . The subgroup structure of Z p is particularly simple: all proper subgroups are finite and cyclic, and there is exactly one of order p n for each nonnegative integer n . One may naturally ask the inverse problem: is G a quasicyclic group if its all proper subgroups are finite or cyclic? The literature [1] gives an affirmative answer if G is an infinite solvable group or G is an infinite group with G > G . In this paper, we will prove that G is a quasicyclic group if its 2000 Mathematics Subject Classification. 20E18, 20F16, 20E34. Key words and phrases. Quasicyclic group, hypoinner group, commutator sub group, solvable group. 357 358 D. ZHENG, Y. MA AND H. LIU all proper normal subgroups are finite or cyclic under the condition that G is an infinite solvable group or G is an infinite group properly containing its commutator group G . A characterization of a group from its subgroups is one of the key technical tools in the infinite group theory. Let G be a group and be an absolute property about group G , i.e. whether G has a property is only dependent on the group G itself. G is called a group if G has a property . G is called an inner group if each proper subgroup of G has the property but G itself doesnt. For more information about inner group, we refer to [2]. In the same vein, we give the following definition for convenience of a later description. Definition 1 . A group G is called a hypoinner group if each proper normal subgroup of G has the property but G itself doesnt....
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 Winter '11
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