Results 1 to 10 of 10

Math Help - Centralizer Proof Attempt

  1. #1
    Member
    Joined
    Aug 2008
    Posts
    98
    Awards
    1

    Centralizer Proof Attempt

    H be a subgroup of G. Show C(C(C(H)))) = C(H)

    hint: show if A is a subgroup of B then C(A) is a subgroup of C(B).

    I started off trying to prove the hint.

    If A is a subgroup of B then I let there intersection be K and then C(K) has only elements of C(A) but they are also contained in C(B) therefor C(A) is a subgropu of C(B).

    Not sure if this is correct and if it is not sure how to apply this to the main question.

    Any help greatly apreciated
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Apr 2009
    Posts
    677
    Quote Originally Posted by Niall101 View Post
    H be a subgroup of G. Show C(C(C(H)))) = C(H)

    hint: show if A is a subgroup of B then C(A) is a subgroup of C(B).

    I started off trying to prove the hint.

    If A is a subgroup of B then I let there intersection be K and then C(K) has only elements of C(A) but they are also contained in C(B) therefor C(A) is a subgropu of C(B).

    Not sure if this is correct and if it is not sure how to apply this to the main question.

    Any help greatly apreciated
    Hint: I guess this should work

    Let x \in C(H) i.e.  xh=hx for all h \in H

    Consider any y \in C(C(H)). Then yc=cy for all c \in C(H). Specifically yx=xy for any y \in C(C(H)). Thus x \in C(C(C(H)))

    I think the reverse will work out similarly.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by Niall101 View Post
    H be a subgroup of G. Show C(C(C(H)))) = C(H)

    hint: show if A is a subgroup of B then C(A) is a subgroup of C(B).

    I started off trying to prove the hint.

    If A is a subgroup of B then I let there intersection be K and then C(K) has only elements of C(A) but they are also contained in C(B) therefor C(A) is a subgropu of C(B).

    Not sure if this is correct and if it is not sure how to apply this to the main question.

    Any help greatly apreciated

    If by C(A) you mean the centralizer in G of the subgroup A then it is false that A\subset B \Longrightarrow C(A)\subset C(B), and a simple example: In\,\,S_3\,,\,\,<(12)>\, \leq S_3\,,\,\,but\,\, C(<(12)>)=<(12)>\nsubseteq \{(1)\}=C(S_3)

    Tonio
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Aug 2008
    Posts
    98
    Awards
    1
    thanks! so basically the hint given is not needed?

    would this result hold for the normalizer of H?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by Niall101 View Post
    thanks! so basically the hint given is not needed?


    Much worse: the hint given is completely false.


    would this result hold for the normalizer of H?

    Nop, that's also false.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by Niall101 View Post
    thanks! so basically the hint given is not needed?


    Much worse: the hint given is completely false.


    would this result hold for the normalizer of H?

    Nop, that's also false.

    Tonio
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member
    Joined
    Apr 2009
    Posts
    677
    Quote Originally Posted by Niall101 View Post
    thanks! so basically the hint given is not needed?

    would this result hold for the normalizer of H?
    In your original problem, after Tonio's quote, I am trying to prove the reverse (without using the hint). But have not been able to do so as yet.
    Last edited by aman_cc; October 29th 2009 at 09:55 AM.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Member
    Joined
    Aug 2008
    Posts
    98
    Awards
    1
    Quote Originally Posted by Niall101 View Post
    H be a subgroup of G. Show C(C(C(H)))) = C(H)

    hint: show if A is a subgroup of B then C(A) is a subgroup of C(B).

    I started off trying to prove the hint.

    If A is a subgroup of B then I let there intersection be K and then C(K) has only elements of C(A) but they are also contained in C(B) therefor C(B) is a subgropu of C(A).

    Not sure if this is correct and if it is not sure how to apply this to the main question.

    Any help greatly apreciated
    My Mistake on the first post
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Super Member
    Joined
    Apr 2009
    Posts
    677
    Quote Originally Posted by aman_cc View Post
    In your original problem, after Tonio's quote, I am trying to prove the reverse (without using the hint). But have not been able to do so as yet.
    Is it correct to say the following:
    If y \in C(C(H)) => y \in C(H)?

    Here is the argument -
    yz=zy for all z \in C(H)
    z^{-1}h=hz^{-1} for all h \in H
    Thus, yh=hy h \in H

    Sorry - Please ignore - This is wrong !
    Follow Math Help Forum on Facebook and Google+

  10. #10
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by Niall101 View Post
    My Mistake on the first post

    It still makes no sense, and the hint remains and remains false: if A is a sbgp. of B, then their interesection is A itself...not very useful, is it?

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Centralizer proof
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 21st 2010, 09:46 PM
  2. verifying identities attempt 3...
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: October 24th 2009, 12:10 AM
  3. one more attempt
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: October 13th 2009, 02:26 PM
  4. Show centralizer of a = centralizer of a^-1
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 25th 2008, 06:47 AM
  5. Attempt to Define Region
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: September 7th 2006, 04:04 AM

Search Tags


/mathhelpforum @mathhelpforum