Results 1 to 2 of 2

Math Help - Lagrange's Thm and normal subgroup

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    83

    Lagrange's Thm and normal subgroup

    If H is a subgroup of a group G, we define:
    G/H = {xH : x belongs to G}
    H\G = {Hx : x belongs to G}
    Let G = D10 = {x, y| x^5 = y^2 = 1, yx =x^4(y)}
    and K = {1, x, x^2,x^3,x^4} and H ={1,y}

    How do you find G/K? and find the normal subgroup of D10

    Can you give me some hints how to do this question plz?

    Cheers
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517

    Cosets

    To find G/K one needs to just consider the cosets. By Lagrange you know that |G/K|=\frac{|G|}{|K|}=\frac{10}{5}=2 so in this case we should expect 2 distinct cosets. One is always 1K and this will be the same coset as when you translate K by anything in K. So to find the other coset try transalating (multiplying on the left) by one of the other elements in D_{10} - K and see what happens.

    Also notice why this particular subgroup is always Normal, and see that there is nothing particularly special about this group, but more importantly that it has size one half as big as the mother group. In fact this is always the case that every subgroup of index 2 is normal, and in more generality if p is the smallest prime that divides the order of the group, any subgroup of index p is normal (note Lagrange does not guarantee the existance of such a subgroup). But in the Dihedral group of order 2n there always is one of index 2 because there is always a cyclic group of order n (the group of rotations of the n-gon in this case the regular pentagon)

    Hope this helps.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: March 2nd 2011, 09:07 PM
  2. Lagrange Theorem & order of subgroup question
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: December 6th 2010, 05:13 AM
  3. Subgroup of cyclic normal subgroup is normal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 25th 2010, 07:13 PM
  4. characterisitic subgroup implies normal subgroup
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 8th 2010, 04:13 PM
  5. Normal subgroup interset Sylow subgroup
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 10th 2008, 01:21 AM

Search Tags


/mathhelpforum @mathhelpforum