cayleyn

Cayley's theorem is a fundamental result in group theory, named after the English mathematician Arthur Cayley. It states that every group is isomorphic to a permutation group. In other words, for any group G, there exists a group of permutations of some set that is structurally identical to G. This theorem is crucial in understanding the relationship between abstract groups and concrete groups of permutations.

The proof of Cayley's theorem involves constructing a permutation representation of the group. For each element g in the group G, a permutation σ_g is defined on the set G itself, where σ_g(h) = gh for all h in G. This mapping g → σ_g defines a homomorphism from G to the group of permutations of G. The kernel of this homomorphism is the identity element, ensuring that the homomorphism is injective. Therefore, G is isomorphic to its image under this homomorphism, which is a group of permutations.

Cayley's theorem has significant implications in group theory and its applications. It provides a way to study any group by considering it as a group of permutations, which can be easier to visualize and manipulate. Additionally, it highlights the importance of permutation groups in the theory of groups. The theorem is also a cornerstone in the classification of finite simple groups, a major achievement in modern mathematics.