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Homework Help: 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

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    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

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    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

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

    Dick

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    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

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

    lanedance

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

    Dick

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    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.
     
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