Vertex groups and bones

That question bugs me for few days. While I filed in searching for the answer, I’m asking here.
Is there a way to connect bone with the given vertex group?

You actually do it the other way round (but only since the Armature is the Modifier) and name your Vertex Group exactly the same as the Bone you want to deform it with.
You can do it the other way round and name the Bone the same as the VG but there are Naming and Syntax conventions for left (.L) and right (.R) used internally and by script authors that will inevitably cause headaches down the line.

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Ahh. Thank you, Fligh!