Results 1 to 2 of 2

Math Help - Determining subgroups

  1. #1
    Newbie
    Joined
    Oct 2007
    Posts
    14

    Determining subgroups

    I am confused as to how I would determine all subgroups of a group. For instance: all the subgroups of the klein-4 group or all the subgroups of the group Z/110z.

    In advance, thanks for the help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by epsilon View Post
    I am confused as to how I would determine all subgroups of a group. For instance: all the subgroups of the klein-4 group or all the subgroups of the group Z/110z.
    I would guess that there is no efficient way of doing this, i.e. brute force is the only way to go. By brute force you form all possible subsets that you can form and see which form a group.

    But pure guessing is not good too you should use some known properties about (finite) groups:
    1)A subgroup must contain the identity element.
    2)The subgroup must divide the order of the group.
    3)The subgroup of a cyclic group is cyclic.

    There might be more well-established facts about subgroups but those three are the most basic ones.

    So in \mathbb{Z}_2\times \mathbb{Z}_2 the Klein-4-group the possible sizes for a subgroup are: 1,2,4. Because those are divisors of 4. 1 being the trivial group. 4 being, the other extreme, the improper group. While the 2 are all intermediate groups. Now it needs to contain #1 which means there are 3 possible elements left to make a 2 element subset. So there are three possibilities to check. Easy.

    With \mathbb{Z}_{110} all subgroups must be cyclic. So create all possible cyclic subgroups and those must be a complete list.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Subgroups and Intersection of Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 1st 2010, 08:12 PM
  2. subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: June 30th 2010, 07:12 PM
  3. subgroups and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 19th 2010, 03:30 PM
  4. Subgroups and Normal Subgroups
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: December 9th 2009, 08:36 AM
  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