A new subgroup lattice characterization of finite solvable groups

Abstract

We show that if G is a finite group then no chain of modular elements in its subgroup lattice L ( G ) is longer than a chief series. Also, we show that if G is a nonsolvable finite group then every maximal chain in L ( G ) has length at least two more than the chief length of G, thereby providing a converse of a result of J. Kohler. Our results enable us to give a new characterization of finite solvable groups involving only the combinatorics of subgroup lattices. Namely, a finite group G is solvable if and only if L ( G ) contains a maximal chain X and a chain M consisting entirely of modular elements, such that X and M have the same length.