Results 1 to 4 of 4

Math Help - Quick questions on Group Theory - Cosets / Normal Group

  1. #1
    Super Member
    Joined
    Apr 2009
    Posts
    677

    Quick questions on Group Theory - Cosets / Normal Group

    Hi - This is just to validate my understanding. You may just say yes/no to questions below (Q4 might need some explanation). If there are deeper concepts involved any pointers would we welcome.

    G be a group. H is a sub-group.
    Let a,b E G

    Hx= Right Coset of H in G, with respect to any x E G
    yH= Left Coset of H in G, with respect to any y E G


    Q1. If Ha = bH, this does not imply aH = bH
    Q2. However if H is Normal then above is true
    Q3. If Ha = aH does not imply Hb = bH

    Q4. If H in not-normal then can this happen Ha = bH (for just some specific a,b and not every a,b)

    Reason why I am asking these - We know well that if every right coset is a left coset; H is Normal and vice-versa.

    I am wondering if H is not normal, can it still happen that some right coset is equal to some left coset? If yes what else can we deduce?

    Coset with respect to 'zero' of the group is trivial so we may exclude that.

    Thanks
    Last edited by aman_cc; October 16th 2009 at 05:22 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by aman_cc View Post
    Hi - This is just to validate my understanding. You may just say yes/no to questions below (Q4 might need some explanation). If there are deeper concepts involved any pointers would we welcome.

    G be a group. H is a sub-group.
    Let a,b E G

    Hx= Right Coset of H in G, with respect to any x E G
    yH= Left Coset of H in G, with respect to any y E G


    Q1. If Ha = bH, this does not imply aH = bH
    this is always true! because if Ha=bH, then a=bh, for some h \in H and thus aH=bhH=bH.


    Q3. If Ha = aH does not imply Hb = bH
    for example let G=S_3 and H=<(2 \ 3)>. let a = (2 \ 3) and b=(1 \ 2).


    Q4. If H in not-normal then can this happen Ha = bH (for just some specific a,b and not every a,b)
    same example of Q3 but this time choose a=(1 \ 2 \ 3), \ b=(1 \ 3 \ 2).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Apr 2009
    Posts
    677
    Quote Originally Posted by NonCommAlg View Post
    this is always true! because if Ha=bH, then a=bh, for some h \in H and thus aH=bhH=bH.




    for example let G=S_3 and H=<(2 \ 3)>. let a = (2 \ 3) and b=(1 \ 2).



    same example of Q3 but this time choose a=(1 \ 2 \ 3), \ b=(1 \ 3 \ 2).
    Thanks. I am sorry for some stupid questions (for e.g. Q1)

    Let me plz state what all I have understood and you can correct it if I am wrong.

    1) H is a normal sub-group IF AND ONLY IF every right coset of H, is a left coset of H as well.

    2) If H is not a normal sub-group then there is atleast one right coset of H which is not a left coset of H

    (I have an unrelated confusion here - if there is one right coset which is not a left coset, does this imply there is atleast one left coset which is not a right coset. Or these two condition are not related?)

    3) ab^{-1} \in H => Ha = Hb; but not aH = bH

    Sorry if I am getting confused un-necessarily plz.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by aman_cc View Post
    Thanks. I am sorry for some stupid questions (for e.g. Q1)

    Let me plz state what all I have understood and you can correct it if I am wrong.

    1) H is a normal sub-group IF AND ONLY IF every right coset of H, is a left coset of H as well.

    2) If H is not a normal sub-group then there is atleast one right coset of H which is not a left coset of H

    (I have an unrelated confusion here - if there is one right coset which is not a left coset, does this imply there is atleast one left coset which is not a right coset. Or these two condition are not related?)

    3) ab^{-1} \in H => Ha = Hb; but not aH = bH

    Sorry if I am getting confused un-necessarily plz.
    the answer to 1) and 2) is yes: H is a normal subgroup if and only if every right (left) coset of H is a left (right) coset of H.

    for 3): we have: ab^{-1} \in H \Longleftrightarrow Ha=Hb. we can have ab^{-1} \in H but aH \neq bH. for example let G=S_3, \ H=<(2 \ 3)>, \ a=(1 \ 2 \ 3), \ b=(1 \ 3).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Transversal And Normal Form Group Theory
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 27th 2011, 07:54 AM
  2. Group Theory - Question on normal subgroup
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: November 10th 2009, 11:44 PM
  3. Replies: 4
    Last Post: August 29th 2009, 11:21 PM
  4. Group Theory, Disjoint cosets dodgy proof problem
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 3rd 2009, 12:45 PM
  5. Group, Cosets, Normal Group help
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: September 10th 2008, 09:12 PM

Search Tags


/mathhelpforum @mathhelpforum