Results 1 to 7 of 7

Math Help - All homomorphisms between S3 and Z6

  1. #1
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    558
    Thanks
    2

    All homomorphisms between S3 and Z6

    Can anyone help with the following problem:

    Find all possible homomorphisms between the indicated groups:

     \phi : S_3 \rightarrow Z_6

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

    Thoughts:

    The only normal subgroups of S_3 are {e} and A_3

    Thus (following the case for {e}) we can assume Ker  \phi = {e} for a homomorphism  \phi (by the First Isomorphism Theorem)

    But how do we find the find the homomorphism - or homomorphisms?

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

    If I can see the situation for {e} then I am hoping a similar analysis will yield the homomorphism(s) for the case of  A_3

    Be grateful for some help.

    Peter
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,273
    Thanks
    666

    Re: All homomorphisms between S3 and Z6

    if ker(φ) = {e}, φ must be injective, which would imply S3 and Z6 were isomorphic. why can't this be the case?

    suppose ker(φ) = A3. then φ(S3) must be isomorphic to S3/A3, which has order 2. what are all the subgroups of Z6 of order 2?

    can you see a possible way to define this homomorphism, based on the parity (even/odd) of an element in S3?

    finally, you are overlooking another possiblity for ker(φ), which produces a rather dull homomorphism.....
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    558
    Thanks
    2

    Re: All homomorphisms between S3 and Z6

    Thanks!!

    I was previously trying to construct a homomorphism between  S_3 and  Z_6 , but given your post I have concluded that this is not possble since by the First Isomorphism Theorem a homomorphism between  S_3 and  Z_6 would imply  S_3  \simeq  Z_6 which is not possible since  Z_6 is cyclic and  S_3 is not.

    Therefore there cannot be a homomorphism between  S_3 and  Z_6 .

    Correct?

    Will now work on the Ker  \phi situation.

    Thanks again for your help.

    Peter
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,273
    Thanks
    666

    Re: All homomorphisms between S3 and Z6

    homomorphism ≠ isomorphism. there cannot be an isomorphism between S3 and Z6 (because why? this is not a hard question....)

    but this doesn't mean there cannot be any homomorphisms between S3 and Z6.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    558
    Thanks
    2

    Re: All homomorphisms between S3 and Z6

    Either I am expressing myself badly or I am wrong - both of which bother me!! :-)

    I was trying to argue thus:

    The First Isomorphism Theorem states that if  \phi : G  \rightarrow G' is a homomorphism with kernel Ker  \phi = K, then G/K  \cong  \phi (G)

    So in the case of  S_3 and  Z_6 where K = {e} we have that  S_3 /{e}  \cong  S_3  \cong  Z_6

    But his cannot be the case ... so therefore there is no homomorphism between  S_3 and  Z_6

    Is that a valid argument?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor

    Joined
    Mar 2011
    From
    Tejas
    Posts
    3,273
    Thanks
    666

    Re: All homomorphisms between S3 and Z6

    no, it only shows that there is no homomorphism with kernel {e}. it does not rule out other possible homomorphisms with different kernels.

    now, WHY can't it be the case that there is no homomorphism with kernel {e}? that is, why aren't S3 and Z6 isomorphic?

    (they both have order 6....they both have subgroups of order 2 and 3.....so what is it, exactly?)
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Super Member Bernhard's Avatar
    Joined
    Jan 2010
    From
    Hobart, Tasmania, Australia
    Posts
    558
    Thanks
    2

    Re: All homomorphisms between S3 and Z6

    In answer to "why aren't  S_3 and  Z_6 isomorphic" it is because  Z_6 is cyclic and  S_3 is not. Therefore there is no homomorphism with kernal {e}.

    Sorry, my assertion that therefore there was no homomorphisms (at all) between  S_3 and  Z_6 was a brain fade!

    Obviously there is the case of the subgroup  A_3 and  S_3 to investigate!

    Still have to work on this one.

    Peter
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 5th 2011, 06:19 PM
  2. Homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: February 6th 2011, 11:39 AM
  3. No of Homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 29th 2010, 04:01 AM
  4. Homomorphisms to and onto
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 23rd 2009, 09:21 AM
  5. Homomorphisms
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: April 23rd 2007, 11:53 AM

Search Tags


/mathhelpforum @mathhelpforum