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On finitary series of groups represented as permutation or linear groups. (English) Zbl 1481.20004

A group \(G\) is called an FC-group if the classes of conjugate elements of \(G\) are finite. The notion of an FC-group was introduced by R. Baer in [Duke Math. J. 15, 1021–1032 (1948; Zbl 0031.19705)]. Then FC-groups were discussed by B. H. Neumann [Proc. Lond. Math. Soc., III. Ser. 1, 178-187 (1951; Zbl 0043.02401)], F. Haimo [Can. J. Math. 5, 498–511 (1953; Zbl 0051.01504)], A. M. Duguid and D. H. McLain [Proc. Camb. Philos. Soc. 52, 391–398 (1956; Zbl 0071.02204] and other authors.
A group \(G\) is called a minimal non-FC-group if \(G\) is not an FC-group, but every proper subgroup of it is an FC-group (\(MN\mathfrak{FC}\), in short). It is not yet known whether there exists a perfect locally finite \(MN\mathfrak{FC}\)-group. This problem has been discussed in [V. V. Belyaev, Sib. Mat. Zh. 19, 509–514 (1978; Zbl 0394.20025); M. Kuzucuoglu and R. E. Phillips, Math. Proc. Camb. Philos. Soc. 105, No. 3, 417–420 (1989; Zbl 0686.20034); F. Leinen, Glasg. Math. J. 41, No. 1, 81–83 (1999; Zbl 0922.20043)]
Let \(G\) be a transitive subgroup of the symmetric group \(\mathrm{Sym}(\Omega)\) on an infinite set \(\Omega\). If for every proper subgroup \(K\) of \(G\) the \(K\)-orbit \(\alpha^K\) is finite for every \(\alpha\in\Omega\), then \(G\) is called a barely transitive permutation group. A permutation group is cofinitary if any nonidentity element fixes only finitely many points.
In the paper under review, the authors note a following problem: Does every perfect locally finite barely transitive group (LFBT, in short) have a nontrivial epimorphic image which is an \(MN\mathfrak{FC}\)-group? The authors give a partial answer to this problem. We give here the formulation of only two results, which do not require additional definitions:
1. (Theorem 1.5) There does not exist a perfect cofinitary LFBT-group.
2. (Theorem 1.6) Let \(p\) be a prime and let \(G\) be a perfect LFBT-\(p\)-group not an \(MN\mathfrak{FC}\)-group. Then every non-trivial element \(g\) of \(G\) can be considered as an invertible linear transformation of some vector space \(V\) such that \(g\) moves and fixes infinitely many elements of \(V\).

MSC:

20B07 General theory for infinite permutation groups
20C32 Representations of infinite symmetric groups
20E25 Local properties of groups
20F24 FC-groups and their generalizations
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References:

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