Results 1 to 5 of 5

Math Help - Subset of all order 2 elements of D_n not a subgroup of D_n

  1. #1
    Member
    Joined
    Jul 2007
    Posts
    90

    Subset of all order 2 elements of D_n not a subgroup of D_n

    For some reason I'm having trouble with this simple exercise. I'm sure I'm simply overlooking something. Thanks.

    Show that \{x \in D_n | x^2=e\} is not a subgroup of D_n (n\geq 3) .
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jul 2007
    Posts
    90
    Ok... so my title is wrong... it's that union e...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by rualin View Post
    For some reason I'm having trouble with this simple exercise. I'm sure I'm simply overlooking something. Thanks.

    Show that \{x \in D_n | x^2=e\} is not a subgroup of D_n (n\geq 3) .
    I believe that set is precisely the reflections plus the identity, plus the 180 degree rotation if n is even, and closure does not hold.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jul 2007
    Posts
    90
    It's reflections, the identity, and if n is even, the rotation by 180 degrees. It is obvious to me that the subset is not a subgroup because it is not closed but I just can't seem to be able to prove it formally.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor undefined's Avatar
    Joined
    Mar 2010
    From
    Chicago
    Posts
    2,340
    Awards
    1
    Quote Originally Posted by rualin View Post
    It's reflections, the identity, and if n is even, the rotation by 180 degrees. It is obvious to me that the subset is not a subgroup because it is not closed but I just can't seem to be able to prove it formally.
    The composition of two reflections is a rotation. When you perform one reflection, the only way to get back to the original orientation via reflection is to do the same reflection again. Thus, since you have at least two (actually 3) reflections in your set, choosing any two of them will result in a rotation that is not the identity, and thus not in the set.

    Edit: Sorry forgot about the 180 degree rotation again... I'm rethinking

    Edit 2: Ah we can just use that S_iS_j=R_{i-j}

    http://en.wikipedia.org/wiki/Dihedra...roup_structure
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Order of elements in a group
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 3rd 2010, 11:51 AM
  2. Prove that a group of order 375 has a subgroup of order 15
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 13th 2010, 11:08 PM
  3. Replies: 1
    Last Post: March 17th 2010, 06:10 PM
  4. Replies: 3
    Last Post: October 11th 2009, 07:03 AM
  5. order of elements
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 27th 2007, 07:56 PM

Search Tags


/mathhelpforum @mathhelpforum