6.11 (1978)

Let $\mathcal{L}$ be the class of locally compact groups with no small subgroups (see D. Montgomery, L. Zippin, Topological transformation groups, New York, 1955; V. M. Glushkov, Uspekhi Matem. Nauk, 12, no. 2 (1957), 3–41 (Russian)). Study extensions of groups in this class with the objective of giving a direct proof of the following: For each $G \in \mathcal{L}$ there exists an $H \in \mathcal{L}$ and a (continuous) homomorphism $\vartheta : G \to H$ with a discrete kernel and an image $\operatorname{Im} \vartheta$ satisfying $\operatorname{Im} \vartheta \cap Z(H) = 1$ ($Z(H)$ is the center of $H$).

This result follows from the Gleason–Montgomery–Zippin solution of Hilbert’s 5th Problem (since $\mathcal{L}$ is the class of finite-dimensional Lie groups and the latter are locally linear). On the other hand a direct proof of this result would give a substantially shorter proof of the 5th Problem since the adjoint representation of $H$ is faithful on $\operatorname{Im} \vartheta$.