Arthur Cayley classified all groups of order $4$ and $6$ in 1854, and groups of order $8$ in 1858. What about groups of order $2,3,5,7$?. These are prime numbers, and the most basic theorem in group theorem, namely Lagrange's theorem, implies that groups of prime order are cyclic.
Today, we know that for every $n\geq 1$, there is only one cyclic group of order $n$, up to isomorphism. (Also, there is only one cyclic group of infinite order.)
Had Arthur Cayley know about classification of finite cyclic groups? Can anyone give a historical information about the "classification of cyclic group"?