On the length and depth of finite groups (with an appendix by D.R. Heath-Brown)

Abstract

An unrefinable chain of a finite group $G$ is a chain of subgroups $G=G_0>G_1>\cdots >G_t=1$, where each $G_i$ is a maximal subgroup of $G_{i-1}$. The length (respectively, depth) of $G$ is the maximal (respectively, minimal) length of such a chain. We studied the depth of finite simple groups in a previous paper, which included a classification of the simple groups of depth $3$. Here we go much further by determining the finite groups of depth $3$ and $4$. We also obtain several new results on the lengths of finite groups. For example, we classify the simple groups of length at most $9$, which extends earlier work of Janko and Harada from the 1960s, and we use this to describe the structure of arbitrary finite groups of small length. We also present a number-theoretic result of Heath-Brown, which implies that there are infinitely many non-abelian simple groups of length at most $9$. Finally we study the chain difference of $G$ (namely the length minus the depth). We obtain results on groups with chain difference $1$ and $2$, including a complete classification of the simple groups with chain difference $2$, extending earlier work of Brewster et al. We also derive a best possible lower bound on the chain ratio (the length divided by the depth) of simple groups, which yields an explicit linear bound on the length of $G/R(G)$ in terms of the chain difference of $G$, where $R(G)$ is the soluble radical of $G$.

29 pages; to appear in Proc. London Math. Soc

References in corpus (2)

• Alternating Subgroups of Exceptional Groups of Lie Type
• The depth of a finite simple group

Cited by in corpus (2)

• Groups with bounded centralizer chains and the~Borovik--Khukhro conjecture
• The length and depth of associative algebras