Question
The set of all bijective functions from a finite set A
to itself forms: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.
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