Is every locally compact, Hausdorff, locally path-connected topological group $G$ locally Euclidean? (That would imply of course also being a Lie group.) Is it true when countable basis is assumed? I wasn't able to find a discussion of this question in the literature on topological groups and the Hilbert 5th problem.


EDIT: As YCor rightly [pointed out](https://mathoverflow.net/questions/385829/are-locally-compact-hausdorff-locally-path-connected-topological-groups-locall#comment983279_385829), let's assume that $G$ is of finite topological dimension. The general theme of my question is to identify a minimal set of conditions on a topological group making it into a Lie group.



Are locally compact, Hausdorff, locally path-connected topological groups locally Euclidean?
Under the additional assumption of finite topological dimension pointed by YCor in the comments to the OP, the answer is *yes*, see e.g. Theorem 10, pp. 120 of the paper of K. Whittington, *Local connectedness in topological groups*, [Topology and its Applications **180** (2015) 111-123](http://dx.doi.org/10.1016/j.topol.2014.11.006). In fact, according to the same Theorem the hypothesis of local compactness is not needed and follows from the others. The second axiom of countability is also not needed.


**Edit:** in due time, the hypotheses needed for a positive solution to Hilbert's 5th problem when the assumption of finite topological dimension is allowed to fail were a major theme in Per Enflo's PhD thesis (1970).


**Edit 2:** Implication (2) $\Rightarrow$ (3) of Theorem 10 *ibid.* is a literal citation of Theorem 4.10.1, page 185 of the book by D. Montgomery and L. Zippin, *Topological Transformation Groups* (Wiley Interscience, 1955), see Theorem 2 in Moishe Kohan's answer.


