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Math Help - Automorphisms and Permutations

  1. #1
    Super Member Bernhard's Avatar
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    Automorphisms and Permutations

    I am reading Dummit and Foote Section 4.4 on Automorphisms.

    In the second papragraph of the secion Dummit and Foote write:

    "Notice that automorphisms of a group G are, in particular, permutations of the set G so Aut(G) is a subgroup of S_G"

    Intuitively it seems to me, at first reflection, at least, that any permututation of G would be an automorphism of G, but based on D&F's statement this is not the case

    Can anyone with more experience of algebra explain at a broad intuitive level why this is not the case?

    Peter
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Automorphisms and Permutations

    Quote Originally Posted by Bernhard View Post
    I am reading Dummit and Foote Section 4.4 on Automorphisms.

    In the second papragraph of the secion Dummit and Foote write:

    "Notice that automorphisms of a group G are, in particular, permutations of the set G so Aut(G) is a subgroup of S_G"

    Intuitively it seems to me, at first reflection, at least, that any permututation of G would be an automorphism of G, but based on D&F's statement this is not the case

    Can anyone with more experience of algebra explain at a broad intuitive level why this is not the case?

    Peter
    I don't know how much intuition you want. Automorphisms are permutations that must satisfy EXTRA, NON-trivial properties. The extra tells us that every automorphism is, first and foremost, a permutation and the NON tells us that not every permutation satisfies this extra condition. For example, an automorphism \sigma:G\to G fixes 1. This automatically knocks out, best case scenario, an |G|^{\text{th}} of the permutations.
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Automorphisms and Permutations

    Thanks

    Yes, getting the idea

    Peter
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