Lemma 6

Let G be a Lie group.

a) Every decreasing sequence of compact subgroups of G is stationary.

b) Let H and $H'$ be two compact subgroups of G such that $H\subset H'$ and such that there exists an isomorphism (of topological groups) from $H'$ to H. Then $H = H'$.

c) With the relation $t\leq t'$, the set $\mathscr{T}$ is a noetherian ordered set (Theory of Sets, Chap. III, §6, no. 5, text preceding Prop. 7).

a) Let $(H_i)_{i\geq 1}$ be a decreasing sequence of compact subgroups of G; these are Lie subgroups of G (Chap. III, §8, no. 2, Th. 2). The sequence of integers (dim $H_i$)$_{i\geq 1}$ is decreasing, hence stationary, so there exists an integer N such that the subgroups $H_i$ have the same identity component for $i\geq N$. Then the decreasing sequence of positive integers $(H_i: (H_i)_0)_{i\geq N}$ is stationary, so $H_i= H_{i+1}$ for $i$ sufficiently large.

b) Let $f$ be an isomorphism from $H'$ to H. The sequence $(f^n(H))_{n\geq 0}$ is a decreasing sequence of compact subgroups of G, so $f^n(H) =f^{n+1}(H)$ for $n$ sufficiently large, by a). Since $f$ is an isomorphism, this implies that $f(H) = H =f(H')$, so $H = H'$.

c) Let $t, t'\in \mathscr{T}$ be such that $t\leq t'$ and $t'\leq t$. Then, there exist $H,H_1\in t$ and $H',H'_1\in t'$ such that $H\supset H'$ and $H_1\subset H'_1$. Let $g$ and $g'$ be two elements of G such that $H_1=gHg^{-1}$ and $H'_1=g'H'g^{'-1}$; put $u=g^{'-1}g$. Then

by b), this implies that $uHu^{-1}= H$, so $H'= H$ and $t'=t$. Thus, the set $\mathscr{T}$ is ordered, and it is noetherian by a).