Corollary 1

Suppose that G is generated by a compact neighbourhood of $e$.

a) There exists a compact subgroup K of G and positive integers $p$ and $q$ such that G is isomorphic to $\mathbf{R}^p\times \mathbf{Z}^q\times K$.

b) Conversely, let K be a compact group, let $p$ and $q$ be positive integers, and let G be a group isomorphic to $\mathbf{R}^p\times \mathbf{Z}^q\times K$. Then K is the unique maximal compact subgroup of G, and the integers $(p, q)$ are determined in a unique way by G.

Assertion a) follows from prop. 1. Let then K be a compact group, and let $p,q$ be positive integers. Suppose that G is isomorphic to the group $\mathbf{R}^p\times \mathbf{Z}^q\times K$, and identify G with this group. By the canonical projection of G onto $\mathbf{R}^p\times \mathbf{Z}^q$, the image of every compact subgroup of G is a compact subgroup of $\mathbf{R}^p\times \mathbf{Z}^q$, and is therefore reduced to the identity element. Hence $K'\subset K$ and K is the largest compact subgroup of G. The subgroup $\mathbf{R}^p\times K$ is also unique since $\mathbf{R}^p$ is the identity component of $G/K$. In view of TG, VII, p. 13, cor. 3, the integer $p$ is determined in a unique way by G. Since $G/(\mathbf{R}^p\times K)$ is isomorphic to $\mathbf{Z}^q$, the integer $q$ is likewise determined in a unique way by G.