The length and depth of compact Lie groups

Let G be a connected Lie group. An unrefinable chain of G is defined to be a chain of subgroups G=G0>G1>⋯>Gt=1 , where each Gi is a maximal connected subgroup of Gi−1 . In this paper, we introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain, and we establish several new results for compact groups. In particular, we compute the exact length and depth of every compact simple Lie group, and draw conclusions for arbitrary connected compact Lie groups G. We obtain best possible bounds on the length of G in terms of its dimension, and characterize the connected compact Lie groups that have equal length and depth. The latter result generalizes a well known theorem of Iwasawa for finite groups. More generally, we establish a best possible upper bound on dimG′ in terms of the chain difference of G, which is its length minus its depth.