How did Arthur Cayley define groups without referring to the nature of their elements?

In 1854, Arthur Cayley became the first mathematician to explicitly define a group abstractly, shifting the focus away from the specific nature of the elements to the properties of the operation acting upon them.1

Rather than relying on concrete objects, Cayley utilized a set of abstract, meaningless symbols (such as 1, α, β, ...) and defined an operation between them using a table. His approach relied on the following principles:1

• Closure: He explicitly required that the operation be closed, meaning the combination of any two elements must result in another element within the set.1
• Implicit Axioms: While he did not formalize them as modern axioms, he implicitly assumed that the operation was associative and that every element possessed an inverse.1

By focusing solely on these operational rules, Cayley was able to deduce fundamental properties of groups, such as the requirement that for a group with n elements, the n-th power of any element must equal the identity element. This work laid the foundation for the abstract group theory used in modern algebra today.12