Finite groups with \(L\)-free lattices of subgroups.
type: Item
instance of: scholarly article
title: Finite groups with \(L\)-free lattices of subgroups.
zbMATH Open document ID: 1193.20018
author: Czesław Bagiński
author: Agnieszka Stocka
published in: Illinois Journal of Mathematics
publication date: 2009-10-15T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ijm/1254403720
review text: The authors study finite groups \(G\) whose subgroup lattices \(L(G)\) are \(L\)-free for certain lattices \(L\), that is, for which \(L(G)\) has no sublattice isomorphic to \(L\). This kind of problem was introduced by them in 1999 when they determined all finite groups \(G\) with \(L(G)\) strongly balanced, which is equivalent to \(L(G)\) being \(L_6\)-free and \(L_7\)-free for certain lattices \(L_6\) and \(L_7\) with 6 and 7 elements, respectively. Later on problems of this type were studied more systematically by the reviewer [Ill. J. Math. 47, No. 1-2, 515-528 (2003; Zbl 1053.20021); J. Group Theory 10, No. 5, 613-631 (2007; Zbl 1128.20014)]. In the present paper the authors are interested in groups \(G\) with \(L(G)\) dually strongly balanced, that is, \(L(G)\) \(D_6\)-free and \(D_7\)-free where \(D_i\) is the lattice dual to \(L_i\) for \(i=6,7\). In all the cases studied so far, the Sylow subgroups of \(G\) had been modular and hence well-known. This is not so for \(D_6\)- and \(D_7\)-free groups and the authors more or less reduce the problem to determine the structure of these groups to the study of \(p\)-groups.
reviewed by: Roland Schmidt
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 06C05
zbMATH DE Number: 5615674
zbMATH Keywords: subgroup lattices
zbMATH Keywords: lattices of subgroups
zbMATH Keywords: \(L\)-free groups
zbMATH Keywords: strongly balanced lattices
zbMATH Keywords: finite groups
MaRDI profile type: Publication
cites work: Finite groups with globally permutable lattice of subgroups
cites work: Nilpotent Groups in Which the Derived Group Has Two Generators
cites work: Q3913574
cites work: Endliche Gruppen I
cites work: Q3934684
cites work: Q4373716
cites work: Powerful \(p\)-groups. I: Finite groups.
cites work: Subgroup lattices of groups
cites work: \(L\)-free groups.
cites work: Planar subgroup lattices.
cites work: On the occurrence of the complete graph \(K_5\) in the Hasse graph of a finite group.
cites work: Nonsolvable finite groups all of whose local subgroups are solvable
description: scientific article; zbMATH DE number 5615674