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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 p-primary component of Q / Z , that is, the unique maximal p-subgroup 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 n-th complex roots of 1, for all n N . A quasicyclic group is the injective hull of Z /p Z (viewing abelian groups as Z-modules). 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 non-negative 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  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, hypo-inner 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 . In the same vein, we give the following definition for convenience of a later description. Definition 1 . A group G is called a hypo-inner group if each proper normal subgroup of G has the property but G itself doesnt....
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