Results 1 to 3 of 3

Math Help - Subgroup index

  1. #1
    Newbie
    Joined
    Jul 2009
    Posts
    21

    Subgroup index

    Suppose H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of index 4 and that G/(H intersect K) is not cyclic.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Shanks's Avatar
    Joined
    Nov 2009
    From
    BeiJing
    Posts
    374
    To prove H n K is a normal subgroup of index 4, you need to prove HK=G;
    and if it is cyclic, then H=K.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Nov 2008
    Posts
    394
    Quote Originally Posted by bethh View Post
    Suppose H and K are distinct subgroups of G of index 2. Prove that H intersect K is a normal subgroup of index 4 and that G/(H intersect K) is not cyclic.
    \frac{H}{H \cap K} \triangleleft \frac{G}{H \cap K} since its index is 2.

    \frac{K}{H \cap K} \triangleleft \frac{G}{H \cap K} since its index is 2.

    \frac{H}{H \cap K} \cap \frac{K}{H \cap K}=1.

    By second isomorphism theorem, \frac{H}{H \cap K} \cong \frac{HK}{K} \cong \frac{G}{K} \cong \mathbb{Z}_2. Similary, \frac{K}{H \cap K} \cong \mathbb{Z}_2.

    It follows that \frac{G}{H \cap K} \cong \frac{H}{H \cap K} \oplus \frac{K}{H \cap K} \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2 , which is isomorphic to Klein-4 group. Thus \frac{G}{H \cap K} is not cyclic.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Normal Subgroup with Prime Index
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 8th 2010, 07:45 PM
  2. Minor Confusion Over Subgroup Index
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 14th 2010, 09:31 PM
  3. Subgroup of index 2 is normal
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: November 7th 2009, 04:08 PM
  4. Subgroup with prime index
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: March 29th 2009, 04:19 PM
  5. Group, subgroup H of index n that is not normal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: November 25th 2008, 05:29 AM

Search Tags


/mathhelpforum @mathhelpforum