1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Feasibility of groups as union of subgroups.

  1. Aug 3, 2010 #1
    1. The problem statement, all variables and given/known data

    I am trying to solve a question from Abstract Algebra by Hernstein.
    Can any one give me hint regarding the following:
    Show that a group can not be written as union of 2 (proper) subgroups although it is possible to express it as union of 3 subgroups?

    Thanks,



    2. Relevant equations



    3. The attempt at a solution
    I know that the union of 2 subgroups is a group only when one is contained in another or viceversa.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 3, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    say you have G with proper subgroups A & B

    consider some cases
    case 1 - say A is contained in B
    ________but A & B are both proper

    case 2 - say A & B are disjoint
    ________now consider the multiplication ab

    case 3 - say A & B have elements in common other than e
    ________consider the set of elements they share

    those cases should cover the range possibilities
     
    Last edited: Aug 3, 2010
  4. Aug 3, 2010 #3

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If G is the union of two proper subgroups A and B then there is an element a of A that is not in B, and an element b of B that's not in A, right? I think that's the only case you need to consider. For the three subgroup case, just try and think of an example.
     
  5. Aug 3, 2010 #4

    lanedance

    User Avatar
    Homework Helper

    trivial, but what if A is a subgroup of B?
     
  6. Aug 3, 2010 #5

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    If B is proper in G and A is a subgroup of B, then the union of A and B is B. Not G.
     
  7. Aug 3, 2010 #6

    lanedance

    User Avatar
    Homework Helper

    yeah i suppose its really obvious, and the idea you give covers both 2 & 3
     
  8. Aug 3, 2010 #7

    lanedance

    User Avatar
    Homework Helper

    as always, well played sir ;)
     
  9. Aug 3, 2010 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    That's the idea. If I were reading a students solution to this problem I'd be interested in the 'group theory' part. The case splitting would just annoy me.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Feasibility of groups as union of subgroups.
  1. Groups and Subgroups (Replies: 8)

  2. Groups and subgroups (Replies: 4)

  3. Groups and subgroups (Replies: 19)

Loading...