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Groups whose proper subgroups of infinite rank have a permutability transitive relation

  • Adolfo Ballester Bolinches , Maria De Falco , Francesco de Giovanni EMAIL logo and Carmela Musella
Published/Copyright: May 17, 2024
Journal of Group Theory
From the journal Journal of Group Theory Volume 27 Issue 6

Abstract

Let 𝐺 be a group. A subgroup 𝐻 of 𝐺 is called permutable if H ⁢ X = X ⁢ H for all subgroups 𝑋 of 𝐺. Permutability is not in general a transitive relation, and 𝐺 is called a PT -group if, whenever 𝐾 is a permutable subgroup of 𝐺 and 𝐻 is a permutable subgroup of 𝐾, we always have that 𝐻 is permutable in 𝐺. The property PT is not inherited by subgroups, and 𝐺 is called a PT ̄ -group if all its subgroups have the PT -property. We prove that if 𝐺 is a soluble group of infinite rank whose proper subgroups of infinite rank have the PT -property, then 𝐺 is a PT ̄ -group.

Acknowledgements

The last three authors are supported by GNSAGA (INdAM) and are members of AGTA – Advances in Group Theory and Applications (www.advgrouptheory.com).

  1. Communicated by: Evgenii I. Khukhro

References

[1] A. Ballester-Bolinches, J. C. Beidleman, R. Esteban-Romero and V. Pérez-Calabuig, Maximal subgroups and PST-groups, Cent. Eur. J. Math. 11 (2013), no. 6, 1078–1082. 10.2478/s11533-013-0222-zSearch in Google Scholar

[2] A. Ballester-Bolinches, L. A. Kurdachenko, J. Otal and T. Pedraza, Infinite groups with many permutable subgroups, Rev. Mat. Iberoam. 24 (2008), no. 3, 745–764. 10.4171/rmi/555Search in Google Scholar

[3] A. Ballester-Bolinches, L. A. Kurdachenko and T. Pedraza, On periodic radical groups in which permutability is a transitive relation, J. Pure Appl. Algebra 210 (2007), no. 3, 665–671. 10.1016/j.jpaa.2006.11.006Search in Google Scholar

[4] J. C. Beidleman, B. Brewster and D. J. S. Robinson, Criteria for permutability to be transitive in finite groups, J. Algebra 222 (1999), no. 2, 400–412. 10.1006/jabr.1998.7964Search in Google Scholar

[5] M. De Falco, F. de Giovanni and C. Musella, Groups whose proper subgroups of infinite rank have a transitive normality relation, Mediterr. J. Math. 10 (2013), no. 4, 1999–2006. 10.1007/s00009-013-0321-xSearch in Google Scholar

[6] M. De Falco, F. de Giovanni and C. Musella, Groups with normality conditions for subgroups of infinite rank, Publ. Mat. 58 (2014), no. 2, 331–340. 10.5565/PUBLMAT_58214_16Search in Google Scholar

[7] M. De Falco, F. de Giovanni and C. Musella, A note on groups of infinite rank with modular subgroup lattice, Monatsh. Math. 176 (2015), no. 1, 81–86. 10.1007/s00605-014-0610-xSearch in Google Scholar

[8] M. De Falco, F. de Giovanni and C. Musella, Groups in which every normal subgroup of infinite rank has finite index, Southeast Asian Bull. Math. 39 (2015), no. 2, 195–201. Search in Google Scholar

[9] M. De Falco, F. de Giovanni, C. Musella and N. Trabelsi, Groups with restrictions on subgroups of infinite rank, Rev. Mat. Iberoam. 30 (2014), no. 2, 537–550. 10.4171/rmi/792Search in Google Scholar

[10] M. De Falco and C. Musella, Groups with many subgroups having modular subgroup lattice, Ricerche Mat. 51 (2002), no. 2, 241–247. Search in Google Scholar

[11] F. de Giovanni and M. Trombetti, Infinite minimal non-hypercyclic groups, J. Algebra Appl. 14 (2015), no. 10, Article ID 1550143. 10.1142/S0219498815501431Search in Google Scholar

[12] M. R. Dixon, M. J. Evans and H. Smith, Locally (soluble-by-finite) groups with all proper insoluble subgroups of finite rank, Arch. Math. (Basel) 68 (1997), no. 2, 100–109. 10.1007/s000130050037Search in Google Scholar

[13] W. Gaschütz, Gruppen, in denen das Normalteilersein transitiv ist, J. Reine Angew. Math. 198 (1957), 87–92. 10.1515/crll.1957.198.87Search in Google Scholar

[14] B. Hartley, Finite groups of automorphisms of locally soluble groups, J. Algebra 57 (1979), no. 1, 242–257. 10.1016/0021-8693(79)90222-9Search in Google Scholar

[15] F. Menegazzo, Gruppi nei quali la relazione di quasi-normalità è transitiva, Rend. Semin. Mat. Univ. Padova 40 (1968), 347–361. Search in Google Scholar

[16] F. Menegazzo, Gruppi nei quali la relazione di quasi-normalità è transitiva. II, Rend. Semin. Mat. Univ. Padova 42 (1969), 389–399. Search in Google Scholar

[17] D. J. S. Robinson, Groups in which normality is a transitive relation, Proc. Cambridge Philos. Soc. 60 (1964), 21–38. 10.1017/S0305004100037403Search in Google Scholar

[18] D. J. S. Robinson, Finiteness Conditions and generalized Soluble Groups, Ergeb. Math. Grenzgeb. (3) 62, Springer, New York, 1972. 10.1007/978-3-662-07241-7Search in Google Scholar

[19] R. Schmidt, Subgroup Lattices of Groups, De Gruyter Exp. Math. 14, Walter de Gruyter, Berlin, 1994. 10.1515/9783110868647Search in Google Scholar

[20] S. E. Stonehewer, Permutable subgroups of infinite groups, Math. Z. 125 (1972), 1–16. 10.1007/BF01111111Search in Google Scholar

[21] G. Zacher, I gruppi risolubili finiti in cui i sottogruppi di composizione coincidono con i sottogruppi quasi-normali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 37 (1964), 150–154. Search in Google Scholar
Received: 2023-12-30
Revised: 2024-03-05
Published Online: 2024-05-17
Published in Print: 2024-11-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Open mappings of locally compact groups
  3. Groups of profinite type and profinite rigidity
  4. On conjugate separability of nilpotent subgroups
  5. Groups whose proper subgroups of infinite rank have a permutability transitive relation
  6. A generalization of the Brauer–Fowler theorem
  7. On groups with large verbal quotients
  8. Conjugacy class numbers and nilpotent subgroups of finite groups
  9. On the common transversal probability
  10. Classifying primitive solvable permutation groups of rank 5 and 6
  11. Tuple regularity and 𝑘-ultrahomogeneity for finite groups
  12. Skew-morphisms of elementary abelian 𝑝-groups
  13. On algebraic normalisers of maximal tori in simple groups of Lie type
https://doi.org/10.1515/jgth-2023-0296

Articles in the same Issue

  1. Frontmatter
  2. Open mappings of locally compact groups
  3. Groups of profinite type and profinite rigidity
  4. On conjugate separability of nilpotent subgroups
  5. Groups whose proper subgroups of infinite rank have a permutability transitive relation
  6. A generalization of the Brauer–Fowler theorem
  7. On groups with large verbal quotients
  8. Conjugacy class numbers and nilpotent subgroups of finite groups
  9. On the common transversal probability
  10. Classifying primitive solvable permutation groups of rank 5 and 6
  11. Tuple regularity and 𝑘-ultrahomogeneity for finite groups
  12. Skew-morphisms of elementary abelian 𝑝-groups
  13. On algebraic normalisers of maximal tori in simple groups of Lie type
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