Abstract
Let G be a connected Lie group. An unrefinable chain of G is defined to be a chain of subgroups G = G_0> G_1> cdots > G_t = 1, where each G_i is a maximal connected subgroup of G_{i-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 dim G' in terms of the chain difference of G, which is its length minus its depth.
Highlights
The maximum length of a chain of subgroups of a finite group G is called the length of G
In [4], we extended these notions to algebraic groups
We show that the depth of an algebraic group behaves rather differently in positive characteristic
Summary
The maximum length of a chain of subgroups of a finite group G is called the length of G. [4, Theorem 5(iii)] states that the depth of a simple classical type algebraic group tends to infinity with the rank of the group. Theorem 2 Let G be a compact connected Lie group and write z = dim Z (G)0, r = rank(G ). N→∞ dim SOn. In addition, we characterize compact connected Lie groups G of small length as follows. Theorem 5 Let c be a positive integer and let G be a compact connected Lie group satisfying l(G) 5 · 2−3/2(√dim G + c). (ii) Let G be a compact connected Lie group and set z = dim Z (G)0 and G = where the Si are pairwise non-isomorphic simple groups. 2 we prove some results on the subgroup structure of compact Lie groups, in particular determining their maximal connected subgroups.
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