\(M_9\)-free groups. (Q486462)

Many interesting classes of lattices can be characterized by the fact that their members are \(L\)-free for certain lattices \(L\), that is, have no sublattice isomorphic to \(L\). For example, it is well-known that a lattice is modular if and only if it is \(L_5\)-free, where \(L_5\) is the non-modular lattice with 5 elements. Let \(L_{10}\) be the subgroup lattice of the dihedral group of order 8. It is easy to see that for every lattice \(L\) such that \(L_5\leq L\leq L_{10}\), the class of finite \(L\)-free groups (that is, groups with \(L\)-free subgroup lattice) is a lattice-defined class of groups lying between the finite modular groups and the finite groups with modular Sylow subgroups. There are precisely six sublattices \(L\) of \(L_{10}\) such that \(L_5<L<L_{10}\): one with 6 elements, one with 7, two with 8, and two with 9 elements. The case when \(|L|\leq 8\) has been considered in some earlier papers. The authors determine the structure of finite \(L\)-free groups for the 9-element non-modular sublattice \(L\) of \(L_{10}\) having only two antiatoms.