A new subgroup lattice characterization of finite solvable groups

TL;DR

This paper establishes a lattice-theoretic characterization of finite solvable groups by relating the lengths of chains of modular elements and maximal chains in their subgroup lattices.

Contribution

It introduces a new combinatorial criterion involving chains of modular elements in the subgroup lattice to characterize finite solvable groups.

Findings

No chain of modular elements exceeds the length of a chief series.

Non-solvable groups have maximal chains at least two longer than the chief length.

A finite group is solvable if and only if certain chains of modular elements and maximal chains have equal length.

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 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.