Abstract
An unrefinable chain of a finite group G is a chain of subgroups G = G0 >G1 > · · · > Gt = 1, where each Gi is a maximal subgroup of Gi−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 numbertheoretic result of HeathBrown, which implies that there are infinitely many nonabelian 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.
Original language  English 

Pages (fromto)  14641492 
Number of pages  29 
Journal  Proceedings of the London Mathematical Society 
Volume  119 
Issue number  6 
Early online date  13 Jun 2019 
DOIs  
Publication status  Published  Dec 2019 
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Profiles

Dr Tim C Burness
 School of Mathematics  Reader in Pure Mathematics
 Pure Mathematics
 Algebra
Person: Academic , Member