Results 1 to 2 of 2

Math Help - proving in cosets

  1. #1
    Member
    Joined
    Mar 2008
    Posts
    87

    proving in cosets

    I have 2 questions on cosets.

    Let G be a group with subgroup \langle h \rangle.
    Let g \in G \backslash \langle h \rangle.
    If there exists N \lhd G such that [a,h] \notin N, show that g \notin \langle h \rangle N.

    I have tried this problem for a long time, but I still can't get a convinced proof.
    Please give me some hint.


    Another question is if aN=bN, is it true that a=b?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by deniselim17 View Post
    I have 2 questions on cosets.

    Let G be a group with subgroup \langle h \rangle.
    Let g \in G \backslash \langle h \rangle.
    If there exists N \lhd G such that [a,h] \notin N, show that g \notin \langle h \rangle N.

    I have tried this problem for a long time, but I still can't get a convinced proof.
    Please give me some hint.


    I assume your "a" must in fact be "g", so [g,h]\notin N\Longrightarrow g^{-1}h^{-1}gh\notin N , so if g\in \langle h\rangle N then

    g=h^rn\,,\,r\in\mathbb{Z}\,,\,n\in N\Longrightarrow [g,h]=n^{-1}h^{-r}h^{-1}h^rnh=n^{-1}h^{-1}nh=[n,h] , but

    this last rightmost expression is in N (why? Hint: normality), so we get a contradiction.



    Another question is if aN=bN, is it true that a=b?


    Not at all: aN=bN\Longleftrightarrow b^{-1}a\in N and that's all.

    For example, take N=5\mathbb{Z}\,,\,a=2\,,\,b=7\Longrightarrow 2+N=7+N

    Tonio


    .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with cosets
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 18th 2011, 08:12 PM
  2. Cosets
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 6th 2010, 05:26 PM
  3. cosets
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: October 27th 2009, 04:51 PM
  4. Cosets
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 22nd 2009, 03:40 PM
  5. Cosets
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: August 25th 2008, 03:26 PM

Search Tags


/mathhelpforum @mathhelpforum