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On hypercentral subgroups of infinite groups. (English) Zbl 1010.20023

The authors start by quoting some results of R. Baer [Trans. Am. Math. Soc. 75, 20-47 (1953; Zbl 0051.25702)] and of T. A. Peng [J. Algebra 48, 46-56 (1977; Zbl 0363.20020), ibid. 78, 431-436 (1982; Zbl 0505.20012)]. From the authors’ introduction: The aim is to extend the above results to infinite groups. The hypercentre of an arbitrary group will be described in terms of the behaviour of its elements, and Peng’s theorem will be generalized to certain classes of groups satisfying suitable finiteness conditions; as a special case we will obtain in particular that such results hold for groups with Chernikov conjugacy classes. Moreover, it will be proved that some embedding properties pertaining to subgroups are countably recognizable.

MSC:

20F19 Generalizations of solvable and nilpotent groups
20E07 Subgroup theorems; subgroup growth
20E15 Chains and lattices of subgroups, subnormal subgroups
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References:

[1] BAER R.: Group elements of prime power index. Trans. Amer. Math. Soc. 75 (1953), 20-47. · Zbl 0051.25702
[2] BAER R.: Abzählbar erkennbare gruppentheoretische Eigenschaften. Math. Z. 79 (1962), 344-363. · Zbl 0105.25901
[3] HEINEKEN H.-MOHAMED I. J.: A group with trivial centre satisfying the normalizer condition. J. Algebra 10 (1968), 368-376. · Zbl 0167.29001
[4] PENG T. A.: The hypercentre of a finite group. J. Algebra 48 (1977), 46-56. · Zbl 0363.20020
[5] PENG T. A.: Hypercentral subgroups of finite groups. J. Algebra 78 (1982), 431- 436. · Zbl 0505.20012
[6] ROBINSON D. J. S.: Finiteness Conditions and Generalized Soluble Groups. Springer,Berlin, 1972. · Zbl 0243.20033
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