Results 1 to 3 of 3

Math Help - Normal subgroups

  1. #1
    Newbie
    Joined
    Nov 2008
    Posts
    16

    Normal subgroups

    This will be my second post of today. Thanks to all who helped before

    Let G = GLn(R) be the group of all invertible n x n matrices with real entries, and let N = {A exists in G | det(A) > 0}
    Show that N is a normal subgroup of G.

    This is the question. I know a few theorems about normal subgroups, and I think I've made some headway. However, I can't seem to piece it all together.

    Obviously, N is the group of invertible, square matrices with positive determinants. This theorem appears useful to me...

    (a^-1)Na is a subset of N, for all a that exist in G.

    By the invertible matrix theorem, I know that the determinant of a (and a^-1) exist. I also know that the determinant of these two values is not 0. I also know that I'm incredibly tempted to just assume commutativity and swap the equation to a(a^-1)N is a subgroup of N, but that's absolutely useless and you can't assume commutativity. Thus, I am stuck here. This seems like the correct theorem to be using, but I can't seem to put it to use.

    Afterwards, I am to find the quotient group G/N, but I can see this being much more clear once I've solved this issue.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by charleschafsky View Post

    Let G = GLn(R) be the group of all invertible n x n matrices with real entries, and let N = {A exists in G | det(A) > 0}
    Show that N is a normal subgroup of G.
    If n\in N and g\in G then \det (gng^{-1}) = \det (g) \det (n) \det (g)^{-1} = \det (n) > 0 and therefore gng^{-1} \in N.
    This means that N\triangleleft G (you need to also show that N is a subgroup first !)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2008
    Posts
    16
    Determinants, the solution to everything! Thanks!
    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: December 1st 2010, 08:12 PM
  2. subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 19th 2010, 03:30 PM
  3. Subgroups and Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 9th 2009, 08:36 AM
  4. normal subgroups...
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 24th 2009, 07:40 PM
  5. Subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: October 13th 2007, 04:35 PM

Search Tags


/mathhelpforum @mathhelpforum