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      Question

      The set of all bijective functions from a finite set A

      to itself forms:
      A A group under composition Correct Answer Incorrect Answer
      B A vector space Correct Answer Incorrect Answer
      C A ring under addition Correct Answer Incorrect Answer
      D A field Correct Answer Incorrect Answer

      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.
      Hence, this structure satisfies all group axioms.

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