On the History of Finite Abstract Groups

The prototype of the abstract group is the special substitution group which is often called permutation group. One of the most fundamental theorems of abstract group theory is that there is one and only one cyclic group of order g, where g represents an arbitrary positive integer. It was stated by A. Cayley, but since no actual definition of the term abstract group was then known, it is clear that no satisfactory proof thereof could be given at that time.