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On certain characterizations of barely transitive groups. (English) Zbl 1202.20002
Introduction: Let \(G\) be a permutation group on an infinite set \(\Omega\). If \(G\) acts transitively on \(\Omega\) and every orbit of every proper subgroup of \(G\) is finite, then \(G\) is called a barely transitive group. This class of groups was considered for the first time by Hartley in connection with the Heineken-Mohamed groups. It is well-known that \(G\) can be represented as a barely transitive group if and only if \(G\) has a subgroup \(H\) of infinite index such that \(\text{Core}_GH=\bigcap_{g\in G}H^g=\langle 1\rangle\) and \(|M:M\cap H|\) is finite for every proper subgroup \(M\) of \(G\). The subgroup \(H\) is called a point stabilizer in \(G\).
Barely transitive groups are mostly considered in the locally finite case and it is well-known that these groups are locally nilpotent \(p\)-groups for a prime \(p\). In recent years some work has appeared on the non-locally finite case. Almost all such results can be found in the survey article [M. Kuzucuoğlu, Turk. J. Math. 24, No. 3, 273-276 (2000; Zbl 0984.20001)].
In [A. Arikan, Rend. Semin. Mat. Univ. Padova 117, 141-146 (2007; Zbl 1166.20001)] locally graded simple non-periodic barely transitive groups are considered and it is proved that all point stabilizers have trivial Hirsch-Plotkin radical (that is, the maximal locally nilpotent normal subgroup is trivial). In the present paper we give certain characterizations of a perfect simple barely transitive group with a point stabilizer whose Hirsch-Plokin radical is non-trivial.
Let \(\mathbf X\) be a class of groups. If every proper subgroup of a group \(G\) is an \(\mathbf X\)-group but \(G\) itself is not, then \(G\) is called a minimal non-\(\mathbf X\) group.
Our main results are as follows: Theorem 1.1. Let \(G\) be a simple barely transitive group with a point stabilizer \(H\). If the Hirsch-Plotkin radical \(F(H)\) of \(H\) is non-trivial, then \(G\) is a periodic finitely generated minimal non-(locally finite) group. If in addition \(H\) is an FC-group, then \(G\) is a two-generated minimal non FC-group.
Corollary 1.2. Let \(G\) be a perfect barely transitive group with a point stabilizer \(H\). If \(H\) is soluble, then \(G\) is a finitely generated periodic group with every proper subgroup locally finite. If in addition \(H\) is an FC-group, then \(G\) is a two-generated minimal non-FC group.
Corollary 1.3. Let \(G\) be a perfect barely transitive group with a point stabilizer \(H\). If \(H\) is hypercentral, then either \(G\) is locally finite or \(G\) is a finitely generated periodic group with every proper subgroup locally finite. If in addition \(H\) is an FC-group, then \(G\) is a two-generated minimal non-FC group.

20B07 General theory for infinite permutation groups
20F50 Periodic groups; locally finite groups
20F24 FC-groups and their generalizations
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
20B22 Multiply transitive infinite groups
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