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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.

##### MSC:
 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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