Automorphism groups of geometric lattices. (Q1771850)

In the paper automorphism groups of particular lattices are considered. The main feature of the authors' approach is to apply strong theorems for groups and rings to be realized as automorphism groups and endomorphism groups respectively. The following main theorem is proved: Let \(G\) be a group of cardinality \(| G| \leq \lambda \) such that \(\lambda ^{\aleph _0} = \lambda \) if \(2^{\aleph _0} \leq | G| \) and \(\lambda =| G|^{\aleph _0}\) otherwise. Obviously, there is a commutative group ring \(R = \mathbb Z A\) generated by \(| G| \) elements with free additive structure and \(G\) a subgroup of \(\Aut R\). Then there is an \(R\)-module \(M\) of cardinality \(\lambda \) such that its lattice \(L\) of finitely generated submodels has size \(\lambda \) and automorphism group \(\Aut L \simeq G\).