Finite groups with \(L\)-free lattices of subgroups.
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.
scientific article; zbMATH DE number 5615674