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Math Help - Prove Inn(G) is isomorphic to G

  1. #1
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    Prove Inn(G) is isomorphic to G

    Show that for G = S3, Inn(G) G.

    Here is what I have:
    Construct φ: G Inn(G) by φ(a) = ia a G.
    (Prove φ is a homomorphism)
    Pick a G
    Then φ(ab) = iab
    But iab(x) = abx(ab)-1
    = abxb-1a-1
    = a(bxb-1)a-1
    = ia(bxb-1)
    = ia(ib(x))
    So, iab = iaib
    Therefore, φ(ab) = φ(a)φ(b)
    And φ is surjective by the definition of Inn(G)

    Can anyone let me know if I am correct? If not, please help.
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  2. #2
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    Quote Originally Posted by page929 View Post
    Show that for G = S3, Inn(G) G.

    Here is what I have:
    Construct φ: G Inn(G) by φ(a) = ia a G.
    (Prove φ is a homomorphism)
    Pick a G
    Then φ(ab) = iab
    But iab(x) = abx(ab)-1
    = abxb-1a-1
    = a(bxb-1)a-1
    = ia(bxb-1)
    = ia(ib(x))
    So, iab = iaib
    Therefore, φ(ab) = φ(a)φ(b)
    And φ is surjective by the definition of Inn(G)

    Can anyone let me know if I am correct? If not, please help.


    You are correct...so far: you still haven't proved thaty Inn(G)\cong G ...

    Tonio
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  3. #3
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    Tonio -

    Can you help me finish? Where do I go from here?

    Thanks
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  4. #4
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    any surjective homomorphism on a finite set is injective because....
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  5. #5
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    Quote Originally Posted by Deveno View Post
    any surjective homomorphism on a finite set is injective because....

    Any surjective map from a finite set to itself (or to a set which has the same number of elements) is injective,

    but here we still don't know whether |Inn(G)|=|G| ...

    Tonio
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  6. #6
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    Quote Originally Posted by page929 View Post
    Tonio -

    Can you help me finish? Where do I go from here?

    Thanks

    Hints:

    1) First prove in general that for any group G\,,\,\,G/Z(G)\cong Inn(G) , using the same map you did in

    your first message.

    2) Now prove directly that Z(S_3)=\{1\}

    Tonio
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  7. #7
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    yes, i know that he has to show that |Inn(G)| = |G|. and it's probably easier to show that S3/Z(S3) is isomorphic to S3, than to prove ia = ib --> a= b.

    alternatively, he could show that the only inner automorphism equal to 1G, the identity map on G, is ie.
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