Exercise 16, § 1

Let $G$ be a subgroup of $\mathbf{R}^n$; suppose that there exists in $G$ a compact connected subset $K$ such that the affine linear variety generated by $K$ has dimension $p$. Prove then that $G$ contains a vector space of dimension $p$. (Reason by induction on $n$, using Exercise 10 c) of VI, p. 23.)