A new subgroup lattice characterization of finite solvable groups

Details A new subgroup lattice characterization of finite solvable groups A new subgroup lattice characterization of finite solvable groups

2010-11-10

arxiv.org/abs/1011.2503v3

J. Algebra 351 (2012), no. 1, 448-458

Mathematics

Group Theory

15 pages; v2 has minor changes for publication; v3 minor typos fixed

Scientific paper

10.1016/j.jalgebra.2011.10.032

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.

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