Some lattice properties of normal-by-finite subgroups

Maria De Falco; Carmela Musella

Bollettino dell'Unione Matematica Italiana (2003)

  • Volume: 6-B, Issue: 3, page 763-771
  • ISSN: 0392-4041

Abstract

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A subgroup H of a group G is said to be normal-by-finite if the core H G of H in G has finite index in H . It has been proved by Buckley, Lennox, Neumann, Smith and Wiegold that if every subgroup of a group G is normal-by-finite, then G is abelian-by-finite, provided that all its periodic homomorphic images are locally finite. The aim of this article is to describe the structure of groups G for which the partially ordered set nf ⁢ G consisting of all normal-by-finite subgroups satisfies certain relevant properties.

How to cite

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De Falco, Maria, and Musella, Carmela. "Some lattice properties of normal-by-finite subgroups." Bollettino dell'Unione Matematica Italiana 6-B.3 (2003): 763-771. <http://eudml.org/doc/194726>.

@article{DeFalco2003,
abstract = {A subgroup $H$ of a group $G$ is said to be normal-by-finite if the core $H_\{G\}$ of $H$ in $G$ has finite index in $H$. It has been proved by Buckley, Lennox, Neumann, Smith and Wiegold that if every subgroup of a group G is normal-by-finite, then $G$ is abelian-by-finite, provided that all its periodic homomorphic images are locally finite. The aim of this article is to describe the structure of groups G for which the partially ordered set $\text\{nf\}(G)$ consisting of all normal-by-finite subgroups satisfies certain relevant properties.},
author = {De Falco, Maria, Musella, Carmela},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {normal-by-finite subgroups; complemented subgroups; completely factorized groups; -groups; subgroups of finite index},
language = {eng},
month = {10},
number = {3},
pages = {763-771},
publisher = {Unione Matematica Italiana},
title = {Some lattice properties of normal-by-finite subgroups},
url = {http://eudml.org/doc/194726},
volume = {6-B},
year = {2003},
}

TY - JOUR
AU - De Falco, Maria
AU - Musella, Carmela
TI - Some lattice properties of normal-by-finite subgroups
JO - Bollettino dell'Unione Matematica Italiana
DA - 2003/10//
PB - Unione Matematica Italiana
VL - 6-B
IS - 3
SP - 763
EP - 771
AB - A subgroup $H$ of a group $G$ is said to be normal-by-finite if the core $H_{G}$ of $H$ in $G$ has finite index in $H$. It has been proved by Buckley, Lennox, Neumann, Smith and Wiegold that if every subgroup of a group G is normal-by-finite, then $G$ is abelian-by-finite, provided that all its periodic homomorphic images are locally finite. The aim of this article is to describe the structure of groups G for which the partially ordered set $\text{nf}(G)$ consisting of all normal-by-finite subgroups satisfies certain relevant properties.
LA - eng
KW - normal-by-finite subgroups; complemented subgroups; completely factorized groups; -groups; subgroups of finite index
UR - http://eudml.org/doc/194726
ER -

References

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  1. BUCKLEY, J. T.- LENNOX, J.C.- NEUMANN, B.H.- SMITH, H.- WIEGOLD, J., Groups with all subgroups normal-by-finite, J. Austral. Math. Soc. (Ser. A), 59 (1995), 384-398. Zbl0853.20023MR1355229
  2. SMITH, H.- WIEGOLD, J., Locally graded groups with all subgroups normal-by-finite, J. Austral. Math. Soc. (Ser. A), 60 (1996), 222-227. Zbl0855.20028MR1375587
  3. CURZIO, M., Sui gruppi per cui sono riducibili alcuni reticoli di sottogruppi notevoli, Matematiche (Catania), 19 (1964), 1-10. Zbl0166.01805MR172933
  4. DE FALCO, M.- MUSELLA, C., Some lattice properties of nearly normal subgroups, Quaderni Mat., 8 (2001), 87-100. Zbl1021.20017MR1949562
  5. DE FALCO, M.- MUSELLA, C., Groups with complete lattice of nearly normal subgroups, Rev. Mat. Complut., 15 (2002), 343-350. Zbl1021.20018MR1951815
  6. EMALDI, M., Sui gruppi risolubili complementati, Rend. Sem. Mat. Univ. Padova, 42 (1969), 123-128. Zbl0275.20073MR255681
  7. FRANCIOSI, S.- DE GIOVANNI, F., Sui gruppi con l'insieme ordinato dei sottogruppi ascendenti riducibile, Ist. Veneto Sci. Lett. Arti Cl. Sci. Mat. Natur., 139 (1980-81), 199-202. 
  8. NAPOLITANI, F., Sui gruppi risolubili complementati, Rend. Sem. Mat. Univ. Padova, 38 (1967), 118-120. Zbl0183.02702MR227283
  9. RINAURO, S., Some lattice properties of virtually normal subgroups, Note Mat., 10 (1990), 273-278. Zbl0791.20025MR1204206
  10. ROBINSON, D. J. S., A Course in the Theory of Groups, Springer, Berlin (1982). Zbl0483.20001MR648604
  11. SCHMIDT, R., Subgroup Lattices of Groups, de Gruyter, Berlin (1994). Zbl0843.20003MR1292462
  12. SUZUKI, M., On the lattice of subgroups of finite groups, Trans. Amer. Math. Soc., 70 (1951), 345-371. Zbl0043.02502MR39717

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