Locally soluble-by-finite groups with small deviation for non-subnormal subgroups

Leonid A. Kurdachenko; Howard Smith

Commentationes Mathematicae Universitatis Carolinae (2007)

  • Volume: 48, Issue: 1, page 1-7
  • ISSN: 0010-2628

Abstract

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A group G has subnormal deviation at most 1 if, for every descending chain H 0 > H 1 > ⋯ of non-subnormal subgroups of G , for all but finitely many i there is no infinite descending chain of non-subnormal subgroups of G that contain H i + 1 and are contained in H i . This property 𝔓 , say, was investigated in a previous paper by the authors, where soluble groups with 𝔓 and locally nilpotent groups with 𝔓 were effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite groups with 𝔓 are soluble-by-finite and are therefore classified.

How to cite

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Kurdachenko, Leonid A., and Smith, Howard. "Locally soluble-by-finite groups with small deviation for non-subnormal subgroups." Commentationes Mathematicae Universitatis Carolinae 48.1 (2007): 1-7. <http://eudml.org/doc/250235>.

@article{Kurdachenko2007,
abstract = {A group $G$ has subnormal deviation at most $1$ if, for every descending chain $H_\{0\}>H_\{1\}>\dots $ of non-subnormal subgroups of $G$, for all but finitely many $i$ there is no infinite descending chain of non-subnormal subgroups of $G$ that contain $H_\{i+1\}$ and are contained in $H_\{i\}$. This property $\mathfrak \{P\}$, say, was investigated in a previous paper by the authors, where soluble groups with $\mathfrak \{P\}$ and locally nilpotent groups with $\mathfrak \{P\}$ were effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite groups with $\mathfrak \{P\}$ are soluble-by-finite and are therefore classified.},
author = {Kurdachenko, Leonid A., Smith, Howard},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {subnormal subgroups; locally soluble-by-finite groups; subnormal subgroups; locally soluble-by-finite groups; descending chains},
language = {eng},
number = {1},
pages = {1-7},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Locally soluble-by-finite groups with small deviation for non-subnormal subgroups},
url = {http://eudml.org/doc/250235},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Kurdachenko, Leonid A.
AU - Smith, Howard
TI - Locally soluble-by-finite groups with small deviation for non-subnormal subgroups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 1
SP - 1
EP - 7
AB - A group $G$ has subnormal deviation at most $1$ if, for every descending chain $H_{0}>H_{1}>\dots $ of non-subnormal subgroups of $G$, for all but finitely many $i$ there is no infinite descending chain of non-subnormal subgroups of $G$ that contain $H_{i+1}$ and are contained in $H_{i}$. This property $\mathfrak {P}$, say, was investigated in a previous paper by the authors, where soluble groups with $\mathfrak {P}$ and locally nilpotent groups with $\mathfrak {P}$ were effectively classified. The present article affirms a conjecture from that article by showing that locally soluble-by-finite groups with $\mathfrak {P}$ are soluble-by-finite and are therefore classified.
LA - eng
KW - subnormal subgroups; locally soluble-by-finite groups; subnormal subgroups; locally soluble-by-finite groups; descending chains
UR - http://eudml.org/doc/250235
ER -

References

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  1. Dixon M.R., Evans M.J., Smith H., Locally (soluble-by-finite) groups with various restrictions on subgroups of infinite rank, Glasgow J. Math. 47 (2005), 309-317. (2005) Zbl1088.20013MR2203498
  2. Kegel O.H., Wehfritz B.A.F., Locally Finite Groups, North-Holland, Amsterdam-London, 1973. 
  3. Kropholler P., On finitely generated soluble groups with no large wreath product sections, Proc. London Math. Soc. 49 (1984), 155-169. (1984) Zbl0537.20013MR0743376
  4. Kurdachenko L.A., Smith H., Groups with the weak minimal condition for non-subnormal subgroups, Ann. Mat. Pura Appl. (4) 173 (1997), 299-312. (1997) Zbl0939.20040MR1625608
  5. Kurdachenko L.A., Smith H., Groups with the weak minimal condition for non-subnormal subgroups II, Comment. Math. Univ. Carolin. 46 (2005), 601-605. (2005) Zbl1106.20023MR2259493
  6. Kurdachenko L.A., Smith H., Groups with small deviation for non-subnormal subgroups, preprint. MR2506959
  7. Möhres W., Auflösbarkeit von Gruppen, deren Untergruppen alle subnormal sind, Arch. Math. (Basel) 54 (1990), 232-235. (1990) MR1037610
  8. Robinson D.J.S., Finiteness Conditions and Generalized Soluble Groups, 2 vols., Springer, New York-Berlin, 1972. Zbl0243.20033

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