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Math Help - Characteristic subgroups are normal

  1. #1
    Super Member Bernhard's Avatar
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    Characteristic subgroups are normal

    Can anyone please help with the following problem from Dummit and Foote:

    ================================================== =======================

    Prove that characteristic subgroups are normal. Give an example of a normal subgroup that is not characteristic.

    ================================================== =======================

    Peter
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  2. #2
    MHF Contributor Amer's Avatar
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    Re: Characteristic subgroups are normal

    you can check this link there are several examples

    Characteristic subgroup - Wikipedia, the free encyclopedia
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    Re: Characteristic subgroups are normal

    if H is a characteristic subgroup of G, then for every φ in Aut(G), φ(H) = H. in particular, if φ is an inner automorphism, φ(x) = gxg^-1,

    then φ(H) = gHg^-1 = H. since this is true of EVERY inner automorphism (for a characteristic subgroup), H is normal.

    an easy example of a normal subgroup which is NOT characteristic is the subgroup H = {1,r^2,s,r^2s} of D8. it's normal because it is of index two.

    but if φ is the automorphism that sends r-->r and s-->rs, then φ(H) = {1,r^2,rs,r^3s} ≠ H.
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    Super Member Bernhard's Avatar
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    Re: Characteristic subgroups are normal

    Thanks for the help

    Still puzzling over this however.

    We have to show that every characteristic group is normal - so surely we have to consider automorphisms which are not inner automorphisms?

    Can you help?

    Peter
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  5. #5
    MHF Contributor Amer's Avatar
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    Re: Characteristic subgroups are normal

    inner automorphisms is a type of the automorphisms
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    Super Member Bernhard's Avatar
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    Re: Characteristic subgroups are normal

    Yes, agree.

    However the definition of characteristic subgroup in Dummit and Foote (page 135) is as follows:

    Defintion. A sugroup H of a group G is called characteristic in G , denoted H char G if *every* automorphism of G maps H to itself i.e. \phi (H) = H for all \phi \in Aut(G).

    My problem or worry with Deveno's proof is that he seems to prove that \phi(H) = H for all inner automorphisms but ... we need to prove the case for all automorphisms ...

    Am I right ... or perhaps missing something ...

    Can someone please clarify this matter

    Peter
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  7. #7
    MHF Contributor Amer's Avatar
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    Re: Characteristic subgroups are normal

    the image of any normal subgroup in any inner automorphism is the normal subgroup
    let
    H normal in G and \phi inner automorphism

    \phi : G \rightarrow G
    \phi (x) = g x g^{-1}
    for any h in H
    \phi (h) = ghg^{-1} \in H 1-1 onto so \phi will map H to H

    for the characteristic subgroup that holds for any auto should not be inner auto
    i.e for normal subgroup you may find an auto which will not map the normal subgroup to itself
    I hope it is clear
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  8. #8
    Super Member Bernhard's Avatar
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    Re: Characteristic subgroups are normal

    Thanks I have looked at it again and think my logic was muddled

    Deveno showed that if we take an inner automorphism (a particular type of automorphism) then the charactersitic subgrup was normal - so that is the end of the story I think - he does not have to prove anything more

    Peter
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    Re: Characteristic subgroups are normal

    another way of saying this is:

    normal = fixed by inner automorphisms
    characteristic = fixed by inner and outer automorphisms

    which makes it clear that all characteristic subgroups are normal, but not necessarily the other way 'round.
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  10. #10
    Super Member Bernhard's Avatar
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    Re: Characteristic subgroups are normal

    Thanks Deveno

    Yes, that makes it very clear.

    Peter
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    Thumbs up Re: Characteristic subgroups are normal

    Quote Originally Posted by Bernhard View Post
    Can anyone please help with the following problem from Dummit and Foote:

    ================================================== =======================

    Prove that characteristic subgroups are normal. Give an example of a normal subgroup that is not characteristic.

    ================================================== =======================

    Peter
    Said well, support you.

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