Results 1 to 4 of 4

Math Help - Normal Subgroups

  1. #1
    Member
    Joined
    Jul 2011
    Posts
    80

    Normal Subgroups

    Let D8={id,r,r^2,r^3,f,fr,fr^2,fr^3} be the dihedral group of order 8. Let N={id,fr}<=D8 and K={id,r^2,fr,fr^3}<=D8. Show that N is a normal subgroup of K, which is a normal subgroup of D8. Also, show that N is not normal in D8. Help, please!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member ILikeSerena's Avatar
    Joined
    Dec 2011
    Posts
    733
    Thanks
    121

    Re: Normal Subgroups

    Quote Originally Posted by lovesmath View Post
    Let D8={id,r,r^2,r^3,f,fr,fr^2,fr^3} be the dihedral group of order 8. Let N={id,fr}<=D8 and K={id,r^2,fr,fr^3}<=D8. Show that N is a normal subgroup of K, which is a normal subgroup of D8. Also, show that N is not normal in D8. Help, please!
    Hi lovesmath!

    Do you have a definition for a normal subgroup handy?
    I'll give one that is reasonably easy to apply to your problem:
    Definition: A subgroup H of G is normal iff \forall g \in G ~ \forall h \in H: ghg^{-1} \in H.

    Can you apply that definition for H=N and G=K?
    Basically it means checking each combination of elements g and h.
    Since it is trivial if either g or h is the identity, there are only 3 cases to check.


    Alternatively, there is also a proposition that you may have:
    Proposition: The kernel of a homomorphism from G to G' is a normal subgroup of G.

    If you can find such a homomorphism this is less work.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Jul 2011
    Posts
    80

    Re: Normal Subgroups

    Would this suffice to show that N is a normal subgroup of K (the first part of the proof)?

    If N is normal in K, then for any x in K and n in N, there is an n' in N such that xn=n'x. Thus, xnx^-1=n', and therefore, xNx^-1 is contained in N. Conversely, if xNx^-1 is contained in N for all x, then letting x=a, we have aNa^-1 is contained in N or aN is contained in Na. On the other hand, letting x=a^-1, we have a^-1(N)(a^-1)^-1=a^-1(N) is contained in N or Na is contained in aN. Thus, N is normal in K.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member ILikeSerena's Avatar
    Joined
    Dec 2011
    Posts
    733
    Thanks
    121

    Re: Normal Subgroups

    It looks like another way to say what a normal subgroup is.
    However, you will have to apply it to the specific groups and their elements.
    You will need to mention for instance the element "fr" and what happens to it.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subgroups and Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 19th 2012, 06:03 AM
  2. Groups, Subgroups and Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 28th 2012, 10:44 AM
  3. Subgroups and Intersection of Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 1st 2010, 08:12 PM
  4. subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 19th 2010, 03:30 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