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Minimal non-CC-groups. (English) Zbl 0644.20025

A CC-group G is one in which \(G/C_ G(x^ G)\) is a Chernikov group for each \(x\in G\). This extension of the concept of FC-group was first considered by Ya. D. Polovickij [Sib. Mat. Zh. 5, 891-895 (1964; Zbl 0143.038)]. Groups in which each proper subgroup is an FC-group have been considered by V. V. Belyaev [ibid. 19, 509-514 (1978; Zbl 0394.20025)], V. V. Belyaev and N. F. Sesekin [Acta Math. Acad. Sci. Hung. 26, 369-376 (1975; Zbl 0335.20013)] and by B. Bruno and R. E. Phillips [Abst. Pap. Am. Math. Soc. 2, 565 (1980)]. In particular, they classified those minimal non-FC-groups which have a non-trivial finite factor group. The main result here shows that if G has a non-trivial finite or abelian factor group and each proper subgroup is a CC-group then G is a CC-group.
It can then be deduced that a locally graded minimal non-CC-group is countable, locally finite and perfect. Further results on these groups have recently been obtained using results on locally finite simple groups.
Reviewer: M.J.Tomkinson

MSC:

20F24 FC-groups and their generalizations
20F50 Periodic groups; locally finite groups
20E07 Subgroup theorems; subgroup growth
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References:

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