The set of all bijective functions from a finite set A

Solution

The set of all bijective functions (i.e., permutations ) from a finite set A to itself is known as the symmetric group on A, commonly denoted Sₙ if A has n elements. This set forms a group under function composition because:

• Closure : Composition of two bijections is a bijection.
• Associativity : Function composition is associative.
• Identity element : The identity function (maps every element to itself) is a bijection and acts as the identity.
• Inverses : Every bijective function has an inverse, which is also a bijection.