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Locally finite minimal non-FC-groups. (English) Zbl 0686.20034
A group G is a minimal non-FC-group, if every proper subgroup of G is an FC-group (a group with finite conjugacy classes), while G itself is not. V. Belyaev [Sib. Mat. Zh. 19, 509-514 (1978; Zbl 0394.20025)] has obtained a considerable amount of information about locally finite imperfect minimal non-FC-groups, and has shown [6th All-Union Symp. group theory, Cherkassy 1978, 97-102 (1980; Zbl 0454.20042)] that a perfect locally finite minimal non-FC-group is either a p-group or simple. In this paper the latter possibility is ruled out; it is not known whether the former can exist. The proof depends on the classification of finite simple groups, via work of the first author and the reviewer on centralizers in simple locally finite groups [Proc. Lond. Math. Soc., III. Ser. (to appear; Zbl 0682.20020)].
Reviewer: B.Hartley

20F50 Periodic groups; locally finite groups
20F24 FC-groups and their generalizations
20E25 Local properties of groups
20E32 Simple groups
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Full Text: DOI
[1] Unsolved Problems in Group Theory: The Kourovka Notebook. (1983)
[2] Thomas, Arch. Math. (Basel) 41 pp 103– (1983) · Zbl 0518.20039
[3] Belyaev, Proc. All Union Symposium on Group Theory pp 97– (1980)
[4] Kegel, Locally Finite Groups (1973)
[5] Thomas, Arch. Math. (Basel) 40 pp 21– (1983) · Zbl 0535.20016
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