A new subgroup lattice characterization of finite solvable groups

JOURNAL OF ALGEBRA

卷:351

Article

Shareshian, John¹  Woodroofe, Russ¹ 

[1] Washington Univ, Dept Math, St Louis, MO 63130 USA

关键词: Subgroup lattice;    Finite solvable group;   

DOI  :  10.1016/j.jalgebra.2011.10.032

来源: Elsevier

【 摘 要 】

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. (C) 2011 Published by Elsevier Inc.

【 授权许可】

Free