%0 Journal Article
%T A new subgroup lattice characterization of finite solvable groups
%A John Shareshian
%A Russ Woodroofe
%J Mathematics
%D 2010
%I arXiv
%R 10.1016/j.jalgebra.2011.10.032
%X 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 that of 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.
%U http://arxiv.org/abs/1011.2503v3