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Basic group theory

  1. Sep 6, 2008 #1

    tgt

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    In a group G, is it true that <A,B>n<C>=<AuBnC> where A,B and C are sets in G?

    Where <D> denotes the smallest subgroup in G containing the set D.

    Proof
    If g is in <A,B> and g is in <C> then g is capable of being generated by elements in A or B and also elements in C. So g is generated by elements in (AuB)nC. So g is in <(AuB)nC>.

    if g is in <(AuB)nC> then g is in <AuB> and g is in <C> so g is in <AuB>n<C>
     
    Last edited: Sep 6, 2008
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  3. Sep 6, 2008 #2

    CompuChip

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    I don't think that is true. For example, consider the additive groups generated by {3} and by {4} (i.e. 3Z and 4Z).
    They both contain 12, yet the intersection of the generating sets is empty.
     
  4. Sep 6, 2008 #3

    tgt

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    nice one.
     
  5. Sep 6, 2008 #4

    tgt

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    We can say that if A and B are subgroups of G then <A,B>=AuB, right?
     
  6. Sep 7, 2008 #5

    CompuChip

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    Did you check with my example?
    [tex]( \langle 3, 4 \rangle, + ) \neq (3\mathbb{Z} + 4\mathbb{Z}, +)[/tex]
    (in fact, the LHS is a group while the RHS is not).
     
  7. Sep 7, 2008 #6

    tgt

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    I see.
     
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