Euler's Theorem that 1+1/4+1/9+1/16+...1/k^2 +...=(pi^2)/6 says to me that the rationals are not closed under addition in the infinite case. The rationals are closed for a finite set. Many years have passed since I studied group theory. Please elaborate on this distinction
An excellent answer. I feel much better. The book, "Journey through Genius" by William Dunham (Normally a fine expositor; I have three of his books.) should have indicated a limit. Then again, Euler himself may not have had the modern understanding of limit.
Group theory begins during the time of Galois (1811-1832).
But you also need to remember the group theory of the 19th century was different than today.
Back in those days group theory was done in terms of permutation groups.
It was Frobenius who gave the a more modern definition in 1887.
(Which happens to be identical to the old definition as Cayley's theorem demonstrates. Though nicer.)