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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.

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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.

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