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Equivalence relation proof

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data
    Let H and K be subgroups of the group G. Let a,b \in G and define a relation on G by a ~ b if and only if a = hbk for some h \in H and k \in K. Prove that this is an equivalence relation.


    2. Relevant equations
    a = hbk


    3. The attempt at a solution
    The goal is to prove the reflexive, symmetric, and transitive properties of equivalence. I was just hoping someone could help lead me in the right direction of how to start each one. Thanks!
     
  2. jcsd
  3. Mar 14, 2012 #2
    Reflexive means a~a. Can you find an element in H and another in K such that a=h.a.k?
    Symmetric means a~b => b~a. So if a=h.b.k, you need to show b=h'.a.k' for some h' in H and k' in K.
    Just use the definitions....
     
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