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MTH633 - Group Theory

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    zareenZ

    Please share idea

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    zareenZ

    @zareen said in MTH633 Assignment 1 Solution and Discussion:

    Write down the elements of group under composition

    Test 0: Is the operation associative? If not, then you’re done. 𝑇(𝑆)

    is not a group. If the operation is associative, then proceed to…

    Test 1: Does the set 𝑇(𝑆)
    contains an identity? If you have a candidate function in 𝑇(𝑆), then what equations must it satisfy? Does it? If not, then 𝑇(𝑆)

    is not a group. If you do have an identity, then proceed to…

    Test 2: Does every element of 𝑇(𝑆)
    have an inverse? Given an arbitrary function in π‘“βˆˆπ‘‡(𝑆), can you write down its inverse π‘“βˆ’1βˆˆπ‘‡(𝑆)? What equation must 𝑓 and π‘“βˆ’1 satisfy? (Hint: you need the identity function.) If any function fails to have an inverse, then 𝑇(𝑆)

    is not a group. If every function does have an inverse, then…

    Congratulations! Your set 𝑇(𝑆)
    is a group.
    Reff