Results 1 to 2 of 2

Math Help - Subgroups and Intersection of Normal Subgroups

  1. #1
    Senior Member
    Joined
    Apr 2008
    From
    Vermont
    Posts
    318

    Subgroups and Intersection of Normal Subgroups

    Prove that if N is a normal subgroup of G and H is any subgroup of G, then H intersect N is a normal subgroup of H.

    N is normal so gng^-1 is in N for all n in N and g in G.
    H is a subgroup so H is closed, has an identity, and has an inverse.

    I don't know how to apply these definitions when thinking about the intersection
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by kathrynmath View Post
    Prove that if N is a normal subgroup of G and H is any subgroup of G, then H intersect N is a normal subgroup of H.

    N is normal so gng^-1 is in N for all n in N and g in G.
    H is a subgroup so H is closed, has an identity, and has an inverse.

    I don't know how to apply these definitions when thinking about the intersection
    Let h\in H be arbitrary. Note first that h\left(N\cap H\right)=(hN)\cap (hH). To see this merely note that x\in h\left(N\cap H\right)\Leftrightarrow x=hy\text{ }y\in H\cap N\Leftrightarrow x=hy\text{ }y\in N\text{ and }x=hy\text{ }y\in H\Leftrightarrow x\in hN\text{ and }x\in hH. Similarly, \left(N\cap H\right)h=(Nh)\cap (Hh).

    Thus, we note that h\left(N\cap H\right)=(hN)\cap (hH)=(hN)\cap H=(Nh)\cap H=(Nh)\cap (Hh)=(N\cap H)h.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. intersection of subgroups
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 25th 2011, 02:07 AM
  2. H and K normal subgroups, intersection of H and K=<e>
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 3rd 2010, 01:40 AM
  3. subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 19th 2010, 04:30 PM
  4. Subgroups and Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 9th 2009, 09:36 AM
  5. Subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: October 13th 2007, 05:35 PM

Search Tags


/mathhelpforum @mathhelpforum