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Math Help - Cosets

  1. #1
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    Cosets

    Hello,

    I am trying to prove, or disprove by means of a counterexample the two statements.

    For a subgroup H of a group G, with elements a,b \in G, and aH is the coset of H containing a, defined as aH = \{ah\ : h \in H\}

    (1)
    aH=bH implies Ha^{-1} = Hb^{-1}

    (2)
    aH = bH implies Ha = Hb

    I'm not sure where to start on these.

    Thanks!
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  2. #2
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    1.

    aH = bH \iff \{ah | h \in H\} = \{bh | h \in H\} \iff \{(ah)^{-1} | h \in H\} = \{(bh)^{-1} |h \in H\}

    And you should be able to see with further manipulations that the implication is indeed true.


    2.

    Look at the group  S_3 . Can you find 2 elements that show this implication is not true?
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  3. #3
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    Just to confirm, in (1), the next step would be
    \{(ah)^{-1} | h \in H\} = \{(bh)^{-1} |h \in H\}
    then
    \{(h)^{-1}(a)^{-1} | h \in H\} = \{(h)^{-1}(b)^{-1} |h \in H\}
    by properties of inverse, then
    \{(h)(a)^{-1} | h \in H\} = \{(h)(b)^{-1} |h \in H\}
    Since if h \in H, \rightarrow h^{-1} \in H
    Then we have the desired result!

    Thanks!!
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