Results 1 to 4 of 4

Thread: normal subgroups

  1. #1
    Member
    Joined
    Mar 2008
    Posts
    96

    normal subgroups

    I have $\displaystyle N_k$ is a normal subgroup of $\displaystyle K$ while $\displaystyle K$ is a normal subgroup of $\displaystyle G$.
    i.e., $\displaystyle N_k \lhd K \lhd G$.

    Is $\displaystyle N_k$ normal to $\displaystyle G$, i.e., $\displaystyle N_k \lhd G$ ? why?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    i found a counter example after posting my reply..
    but,.. after i got home, alice had posted the counter example i was thinking...
    Last edited by kalagota; Feb 22nd 2009 at 03:13 AM. Reason: error..
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by deniselim17 View Post
    Is $\displaystyle N_k$ normal to $\displaystyle G$, i.e., $\displaystyle N_k \lhd G$ ? why?
    A counterexample is,

    Consider $\displaystyle D_4=\{1, r, r^2, r^3, s, rs, r^2s, r^3s\}$.

    $\displaystyle D_4$ has a subgroup isomorphic to a Klein-4 group, which is $\displaystyle \{1, r^2, s, r^2s\}$.
    A normal subgroup of $\displaystyle \{1, r^2, s, r^2s\}$ is $\displaystyle \{1,s\}$. However, $\displaystyle N=\{1,s\}$ is not normal in $\displaystyle D_4$, since $\displaystyle rN \neq Nr$.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Here is another counterexample.
    Consider $\displaystyle S_4$ and $\displaystyle V = \{ \text{id},(12)(34),(13)(24),(14)(23)\}$ and let $\displaystyle H = \{ \text{id},(12)(34)\}$.
    We see that $\displaystyle V\triangleleft S_4\text{ and }H\triangleleft V$ but $\displaystyle H\not \triangleleft S_4$.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subgroups and Intersection of Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Dec 1st 2010, 08:12 PM
  2. subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Mar 19th 2010, 03:30 PM
  3. Subgroups and Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: Dec 9th 2009, 08:36 AM
  4. Normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Dec 13th 2008, 05:39 PM
  5. Subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: Oct 13th 2007, 04:35 PM

Search Tags


/mathhelpforum @mathhelpforum