I haven't thought too hard about this, so it might be easy. Analysis counterexamples aren't my strong suit.
One slick way of phrasing the definition of a Lie Group is as a group object in the category of smooth manifolds. In particular, this forces the group multiplication to be smooth.
You can also ask for groups in the category of, say, $C^1$ manifolds (or any $C^k$). However we know (by a theorem of Whitney) that any $C^k$ manifold already has smooth structure by restricting the atlas (though the resulting smooth structure may be noncanonical).
Question: Are there groups whose multiplication is $C^1$ but not $C^2$? What about $C^k$ but not $C^{k+1}$?
It seems like such objects should exist, but I haven't encountered them, and I can't find any by a quick google.
Thanks in advance ^_^
