The length and depth of real algebraic groups

2018/11/15 by Damian Sercombe, Sercombe, Damian
Mathematics · #20G20 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20G20

paper · pdf · doi:10.48550/arxiv.1811.06327

arxiv created 2018/11/15 · arxiv updated 2018/11/16

Abstract

Let G be a connected real algebraic group. An unrefinable chain of G is a chain of subgroups G=G₀>G₁>...>G_t=1 where each G_i is a maximal connected real subgroup of G_i-1. The maximal (respectively, minimal) length of such an unrefinable chain is called the length (respectively, depth) of G. We give a precise formula for the length of G, which generalises results of Burness, Liebeck and Shalev on complex algebraic groups and also on compact Lie groups. If G is simple then we bound the depth of G above and below, and in many cases we compute the exact value. In particular, the depth of any simple G is at most 9.

Related

• The length and depth of algebraic groups
• The length and depth of compact Lie groups
• The depth of a finite simple group
• Nonsoluble and non-p-soluble length of finite groups
• On the depth of subgroups of simple groups