Results 1 to 2 of 2

Math Help - groups question

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    51

    groups question

    Can there be a homomophism f, from Z16 onto Z2+Z2?
    To figure this out I tried to use the properties of subgroups of homomorphisms. If H is a subgroup Z16, then f(H) is cyclic. But all of the subgroups of Z2+Z2 are also cyclic so I am having a hard time getting a contradiction. Can anyone explain to me were I am going wrong on this?
    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by wutang View Post
    Can there be a homomophism f, from Z16 onto Z2+Z2?
    To figure this out I tried to use the properties of subgroups of homomorphisms. If H is a subgroup Z16, then f(H) is cyclic. But all of the subgroups of Z2+Z2 are also cyclic so I am having a hard time getting a contradiction. Can anyone explain to me were I am going wrong on this?
    Thanks

    Suppose \phi:\,\mathbb{Z}_{16}\rightarrow \mathbb{Z}_2\times\mathbb{Z}_2 is an epimorphism \Longrightarrow \mathbb{Z}_{16}/\ker(\phi) \cong \mathbb{Z}_2\times \mathbb{Z}_2 , which of course is a contradiction since any homomorphic image of a cyclic

    group is cyclic ( in our case, if \mathbb{Z}_{16}=<a>\,,\,\,then\,\,\,\mathbb{Z}_{16  }/\ker(\phi)=<a\ker(\phi)> )

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Factor groups question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 25th 2011, 03:03 PM
  2. p-Groups Question
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: January 21st 2011, 06:28 AM
  3. question about isomorphic groups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: January 27th 2010, 09:01 AM
  4. Order of groups involving conjugates and abelian groups
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: February 5th 2009, 08:55 PM
  5. Question regarding groups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: August 29th 2007, 10:33 PM

Search Tags


/mathhelpforum @mathhelpforum