The way I was taught was that back in the olden days, "group" always referred to groups of permutations (and the operation was always composition), and it was C

The way I was taught was that back in the olden days, "group" always referred to groups of permutations (and the operation was always composition), and it was Cayley that introduced the much more general and abstract notion of groups that we use now. He could do that because it's relatively trivial to prove that the the two definitions are basically the same: every group is isomorphic to some group of permutations according to Cayley's theorem: https://en.wikipedia.org/wiki/Cayley%27s_theorem https://en.wikipedia.org/wiki/Cayley%27s_theorem