TITLE: Is there a topological group that is connected but not path-connected?
QUESTION [17 upvotes]: Is there a $\big($T$_0$$\hspace{-0.02 in}\big)$ topological group that is connected but not path-connected?
If yes:
$\quad$ Can it be complete? $\:$ (with respect to the two-sided uniform structure)

$\quad$ Can it be abelian?

$\quad$ Can it be abelian and complete? $\:$ (simultaneously)

Searching online for various combinations of "topological group", "connected",

and "path-connected" did not turn up anything related to this question.

REPLY [12 votes]: Yes to all questions (as implicit in the comments, so I put cw). Indeed the solenoid defined as the inverse limit of the sequence of surjective endomorphisms $\mathbf{R}/\mathbf{Z}$ given by multiplication by 2 is a compact, metrizable, connected and not path-connected group.