42(2)-09 - GLASNIK MATEMATI ˇ CKI Vol 42(62(2007 357 –...

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Unformatted text preview: GLASNIK MATEMATI ˇ 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 Pr¨ufer 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 [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, hypo-inner Σ 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 Σ be an absolute property about group G , i.e. whether G has a property Σ is only dependent on the group G itself. G is called a Σ group if G has a property Σ. G is called an inner Σ group if each proper subgroup of G has the property Σ but G itself doesn’t. For more information about inner Σ 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 hypo-inner Σ group if each proper normal subgroup of G has the property Σ but G itself doesn’t....
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42(2)-09 - GLASNIK MATEMATI ˇ CKI Vol 42(62(2007 357 –...

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