Results 1 to 6 of 6

Math Help - Help:--Subgroups

  1. #1
    Member
    Joined
    May 2008
    Posts
    102
    Awards
    1

    Help:--Subgroups

    Hello...
    I need real help in these problems...
    Please help!

    Q1) Prove that if H and K are subgroups of a group G (with operation *), then H intersection K is a subgroup of G.

    Q2) Let H = {(1), (1 2)} and K = {(1), (1 2 3), (1 3 2)}. Both H and K are subgroup of S3. Show that HUK is not a subgroup of S3. It follows that a union of subgroup is not necessarily a subgroup.


    Thanks in advance!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,617
    Thanks
    1581
    Awards
    1
    Do you know what property of a set must be shown in order to prove the set is a subgroup? Please post what you know about this.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    May 2008
    Posts
    102
    Awards
    1
    Sir, this is what I could find in the book:

    Let G be a group with operation * and let H be a subset of G. Then H is a subgroup of G if and only if:
    1) H is non empty.

    2) If a E H and b E H then a*b E H

    3) If a E H then a(inverse) E H.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by Vedicmaths View Post
    Hello...
    I need real help in these problems...
    Please help!

    Q1) Prove that if H and K are subgroups of a group G (with operation *), then H intersection K is a subgroup of G.
    you were right in what you said you needed to check if a group is a subgroup of another.

    the first is that it is non-empty.

    well, is the intersection non-empty? is there an element that every subgroup of a group would have to have? if so, that element would be in the intersection of subgroups, right?

    the second is closure under the operation.

    well, is the intersection closed under the operation? take any two elements in the intersection, how would we know whether or not their "product" is in the intersection?

    the last is the presence of inverses for each element. so take an element in the intersection. is its inverse in the intersection? how would you show that?


    Q2) Let H = {(1), (1 2)} and K = {(1), (1 2 3), (1 3 2)}. Both H and K are subgroup of S3. Show that HUK is not a subgroup of S3. It follows that a union of subgroup is not necessarily a subgroup.
    first thing's first. what is the set H \cup K?

    all you need to show is that it violates one of the properties of being a sub-group. also note that subgroups are groups themselves. so if H \cup K is not a group itself, that is also sufficient to prove it is not a subgroup. just go through and check the properties one by one. as soon as you find a violation, stop
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    May 2008
    Posts
    102
    Awards
    1
    1) To show every element in the intersection has an inverse:
    If a is in H ∩ K, then a is in H and a is in K. Since both of them are subgroups, a^-1 is in H and a^-1 is in K. Thus a^-1 is in H ∩ K.

    2) What is (1 2 3) * (1 2)? Is it in H U K?...I thought union of that subgroup is a subgroup. Hence, since HUK is not in there...I don't know how to deal with??
    Follow Math Help Forum on Facebook and Google+

  6. #6
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by Vedicmaths View Post
    1) To show every element in the intersection has an inverse:
    If a is in H ∩ K, then a is in H and a is in K. Since both of them are subgroups, a^-1 is in H and a^-1 is in K. Thus a^-1 is in H ∩ K.
    that is correct. however, the are THREE things you need to show. you only showed one. look back at my (and your) posts. in addition to what you did, you need to show that the intersection is non-empty and that we have closure under the operation of G

    2) What is (1 2 3) * (1 2)? Is it in H U K?...I thought union of that subgroup is a subgroup. Hence, since HUK is not in there...I don't know how to deal with??
    (1 2 3)(1 2) = (1 3) ...what does that tell us?
    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 and normal subgroups
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 19th 2010, 03:30 PM
  3. subgroups
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: March 10th 2010, 09:56 AM
  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