Cayley table in the context of Symmetric group

In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely speaking, the symmetry group of the object.

In mathematics, the special linear group ${\displaystyle \operatorname {SL} (n,R)}$ of degree ${\displaystyle n}$ over a commutative ring ${\displaystyle R}$ is the set of ${\displaystyle n\times n}$ matrices with determinant ${\displaystyle 1}$, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant

where ${\displaystyle R^{\times }}$ is the multiplicative group of ${\displaystyle R}$ (that is, ${\displaystyle R}$ excluding ${\displaystyle 0}$ when ${\displaystyle R}$ is a field).

Arthur Cayley FRS (/ˈkeɪli/; 16 August 1821 – 26 January 1895) was an English mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years.

He postulated what is now known as the Cayley–Hamilton theorem—that every square matrix is a root of its own characteristic polynomial, and verified it for matrices of order 2 and 3. He was the first to define the concept of an abstract group, a set with a binary operation satisfying certain laws, as opposed to Évariste Galois' concept of permutation groups. In group theory, Cayley tables, Cayley graphs, and Cayley's theorem are named in his honour, as well as Cayley's formula in combinatorics.