[bkovitz]

Actually, I believe group theory emerged very gradually from many simultaneous developments. Other sources: permutations, symmetries of crystals and linear transformations, quaternions and matrices (though theory of equations was indeed the main driver). I believe it wasn't until after a bunch of these had reached a high level of development (around 1880) that the abstract group concept began attracting attention as a field of research, because then people could see its use for unifying all those others. Abel's proof for 5th-degree polynomials came out in 1824, and Galois' general proof was published in 1846, so this was slow going, even by nineteenth-century standards.

BTW, I thought I read that Gödel worked pretty much as a lone wolf on his stuff, as most mathematicians weren't into it. And the trend has continued: foundations of mathematics has gotten research attention from surprisingly few people. I'm not 100% sure of this, though.

But anyway, yeah, proving that something can't be done is definitely a big deal. Interesting observation. I will percolate on that.