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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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