Results 1 to 3 of 3

Math Help - normal subgroup proof

  1. #1
    Junior Member platinumpimp68plus1's Avatar
    Joined
    Dec 2008
    Posts
    38

    normal subgroup proof

    If H is a normal subgroup of G, and (G:H)=m, show that a^m is in H for all a in G.

    i know this means that G/H has order m. i started the proof as follows:

    a^mH=(aH)(aH)...(aH) (m times)

    aH must have order less than or equal to m. if its equal to m, then it gives back the identity in G/H (H), which means a^m=h for some h, and is therefore in H, and the proof is done.

    BUT does the order of aH necessarily have to be m?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by platinumpimp68plus1 View Post
    If H is a normal subgroup of G, and (G:H)=m, show that a^m is in H for all a in G.

    i know this means that G/H has order m. i started the proof as follows:

    a^mH=(aH)(aH)...(aH) (m times)

    aH must have order less than or equal to m. if its equal to m, then it gives back the identity in G/H (H), which means a^m=h for some h, and is therefore in H, and the proof is done.

    BUT does the order of aH necessarily have to be m?

    Hint: if ord(G)=n for some group, then g^m=1\,\,\,\forall g\in G

    Tonio
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member platinumpimp68plus1's Avatar
    Joined
    Dec 2008
    Posts
    38
    oooh right. by lagrange the order of the element has to divide the order of the subgroup... so a^m H=eH whether its order is exactly m or not, ie. a^m is in the identity of G/H, which is exactly H. thanks.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Herstein: Normal subgroup proof.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 16th 2011, 04:07 PM
  2. Replies: 2
    Last Post: March 2nd 2011, 09:07 PM
  3. normal subgroup proof
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: October 17th 2010, 07:06 PM
  4. Urgent - normal subgroup proof
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 16th 2009, 02:28 AM
  5. Factor group/Normal subgroup proof
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: April 29th 2007, 07:50 PM

Search Tags


/mathhelpforum @mathhelpforum