Corollary

— Let G be a solvable group of continuous affine transformations on K. Then there exists a point in K which is invariant under G.

By the definition of a solvable group (A, I, § 6, No. 4) there exists a finite decreasing sequence $(G_i)_{0 \leq i \leq n}$ of distinct subgroups of G, such that $G_0 = G$, $G_n = \{e\}$ and such that the group $G_{i-1}/G_i$ is commutative for $1 \leq i \leq n$. Let $K_i$ denote the set of fixed points of $G_i$ in K. Then $K_n = K$. Moreover, for $1 \leq i \leq n$, every element of $G_i$ induces the identity transformation on $K_i$; we thus deduce an action of the abelian group $G_{i-1}/G_i$ on K if $K_i$ is non-empty; it follows from th. 1 that the set $K_{i-1}$ of fixed points of $G_{i-1}/G_i$ in $K_i$ is non-empty. By descending induction on $i$, we conclude that $K_0$ is non-empty, hence the corollary.