A new subgroup lattice characterization of finite solvable groups

作者：Shareshian John^*; Woodroofe Russ

来源：Journal of Algebra, 2012, 351(1): 448-458.

DOI：10.1016/j.jalgebra.2011.10.032

• Subgroup lattice
• Finite solvable group

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 G(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 G(G) contains a maximal chain X and a chain M consisting entirely of modular elements, such that X and M have the same length.

• 出版日期2012-2-1