On certain characterizations of barely transitive groups.

*(English)*Zbl 1202.20002Introduction: 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.

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 |

##### Keywords:

barely transitive groups; subgroups of infinite index; locally finite groups; point stabilizers; Hirsch-Plotkin radical; locally nilpotent normal subgroups; perfect groups; FC-groups; periodic groups##### References:

[1] | A. Arikan, On barely transitive \(p\) groups with soluble point stabilizer, J. Group Theory, 5, No. 4 (2002), pp. 441–442. Zbl1015.20002 MR1931368 · Zbl 1015.20002 |

[2] | A. Arikan, On locally graded non-periodic barely transitive groups , Rend. Semin. Mat. Univ. Padova, 117 (2007), pp. 141–146. Zbl1166.20001 MR2351790 · Zbl 1166.20001 |

[3] | V. V. Belyaev, Groups of the Miller-Moreno type, (Russian) Sibirsk. Mat. Z., 19, no. 3 (1978), pp. 509–514. Zbl0394.20025 MR577067 · Zbl 0394.20025 |

[4] | V. V. Belyaev, Inert subgroups in infinite simple groups (Russian). Sibirsk. Math. Zh., 34 (1993), pp. 17–23; translation in Siberian Math. J., 34 (1993), pp. 606–611. Zbl0831.20033 MR1248784 · Zbl 0831.20033 |

[5] | V. V. Belyaev - M. Kuzucuoǧlu, Locally finite barely transitive groups (Russian), Algebra i Logika, 42 (2003), pp. 261–270; translation in Algebra and Logic, 42 (2003), pp. 147–152. Zbl1033.20001 MR2000842 · Zbl 1033.20001 |

[6] | M. R. Dixon - M. J. Evans - H. Smith, Groups with all proper subgroups soluble-by-finite rank, J. Algebra, 289 (2005), pp. 135–147. Zbl1083.20034 MR2139095 · Zbl 1083.20034 |

[7] | B. Hartley, On the normalizer condition and barely transitive permutation groups, Algebra and Logic, 13 (1974), pp. 334–340. Zbl0319.20052 MR393243 · Zbl 0319.20052 |

[8] | B. Hartley - M. Kuzucuoǧlu, Non-simplicity of locally finite barely transitive groups, Proc. Edin. Math. Soc., II. Ser., 40, No. 3 (1997), pp. 483–490. Zbl0904.20031 MR1475911 · Zbl 0904.20031 |

[9] | O. H. Kegel - B. A. F. Wehrfritz, Locally Finite Groups, North-Holland Math. Library, vol. 3, North-Holland, Amsterdam, 1973. Zbl0259.20001 MR470081 · Zbl 0259.20001 |

[10] | M. Kuzucuoǧlu - R. E. Phillips, Locally finite minimal non-FC-groups. Math. Proc. Cambridge Philos. Soc., 105, no. 3 (1989), pp. 417–420. Zbl0686.20034 MR985676 · Zbl 0686.20034 |

[11] | M. Kuzucuoǧlu, Barely transitive permutation groups, Arch. Math., 55 (1990), pp. 521–530. Zbl0694.20004 MR1078272 · Zbl 0694.20004 |

[12] | M. Kuzucuoǧlu, On torsion free barely transitive groups, Turkish J. Math., 24 (2000), pp. 273–276. Zbl0984.20001 MR1797525 · Zbl 0984.20001 |

[13] | M. Kuzucuoǧlu, Barely transitive groups, Turkish J. Math., 31 (2007), Suppl. pp. 79–94. Zbl1139.20034 MR2368084 · Zbl 1139.20034 |

[14] | D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Vols. 1 and 2, ( Springer -Verlag 1972). Zbl0243.20033 · Zbl 0243.20033 |

[15] | A. I. Sozutov, Residually finite groups with nontrivial intersections of pairs of subgroups, Sibirsk. Mat. Zh., 41, no. 2 (2000), pp. 437–441, iv; translation in Siberian Math. J., 41, no. 2 (2000), pp. 362–365. Zbl0956.20018 MR1762195 · Zbl 0956.20018 |

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