Results 1 to 3 of 3

Thread: Normal subgroup question

  1. #1
    Junior Member
    Joined
    Dec 2009
    Posts
    30

    Normal subgroup question

    Hi, everyone! This is my first post on the forums.

    There is a practice problem I've found that states:

    "Given N1, a normal subgroup of G1, and N2, a normal subgroup of G2, show N1 X N2 is a normal subgroup of G1 X G2."

    Now I've been doing a lot of approaches, but the one that seems to be working out for me hinges on if I can assume that N1 is a normal subgroup of G1 X G2 (which then means I can assume as well that N2 is a normal subgroup of G1 X G2). But can I make this assumption? Is it true? Why or why not?

    EDIT: I realize that I should be working with N1 X Z1 and N2 X Z1 above so that they can be normal subgroups (which are isomorphic to N1 and N2 respectively). I'm still unsure of how to show that it is the case that N1 X Z1 is a normal subgroup of G1 X G2 and N2 X Z1 is a normal subgroup of G1 X G2.

    Thanks for any input!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Nov 2009
    Posts
    169
    Elements of $\displaystyle G_1 \times G_2$ are pairs of elements of G1 and G2, while elements of $\displaystyle N_1 \times N_2$ are pairs of elements of N1 and N2.
    If $\displaystyle g \in G_1 \times G_2$, and $\displaystyle h \in N_1 \times N_2$ how can you write them in terms of each group? How is group multiplication defined in $\displaystyle G_1 \times G_2$? What can you then say about $\displaystyle g^{-1}hg$?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Dec 2009
    Posts
    30
    Here was my outline for doing the whole problem:

    $\displaystyle N_1 \times N_2 \triangleright G_1 \times G_2 \leftrightarrow (g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2$

    So, to show that $\displaystyle N_1 \times N_2 \triangleright G_1 \times G_2$, I want to show that $\displaystyle (g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2$

    [If I had $\displaystyle N_1 \times Z_1 \triangleright G_1 \times G_2$ and $\displaystyle N_1 \times Z_1 \cong N_1$, then $\displaystyle (g_1g_2)(n_1)(g_1g_2)^{-1} \in N_1$]

    Similarly,

    [If I had $\displaystyle N_2 \times Z_1 \triangleright G_1 \times G_2$ and $\displaystyle N_2 \times Z_1 \cong N_2$, then $\displaystyle (g_1g_2)(n_2)(g_1g_2)^{-1} \in N_2$]

    Then by looking at $\displaystyle N_1 \times N_2$, I would multiply the previous results to obtain this:

    $\displaystyle (g_1g_2)(n_1)(g_1g_2)^{-1}(g_1g_2)(n_2)(g_1g_2)^{-1} \in N_1 \times N_2$

    $\displaystyle = (g_1g_2)(n_1)g_2^{-1}g_1^{-1}g_1g_2(n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2$

    $\displaystyle = (g_1g_2)(n_1)g_2^{-1}g_2(n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2$

    $\displaystyle = (g_1g_2)(n_1n_2)g_2^{-1}g_1^{-1} \in N_1 \times N_2$

    $\displaystyle = (g_1g_2)(n_1n_2)(g_1g_2)^{-1} \in N_1 \times N_2$ QED.

    Is this an acceptable approach? The material in the colored brackets is what I'm not comfortable assuming.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: Mar 2nd 2011, 08:07 PM
  2. normal subgroup question
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 23rd 2010, 05:55 PM
  3. normal subgroup question
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Apr 15th 2010, 07:09 PM
  4. Group Theory - Question on normal subgroup
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: Nov 10th 2009, 11:44 PM
  5. Normal subgroup interset Sylow subgroup
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 10th 2008, 12:21 AM

Search tags for this page

Click on a term to search for related topics.

Search Tags


/mathhelpforum @mathhelpforum