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Thread: What does this relation mean?

  1. #1
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    What does this relation mean?

    What does it mean to multiply an element of a group by a subgroup?

    From my Discrete class notes:
    Let $\displaystyle G$ be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let $\displaystyle H$ be a subgroup of $\displaystyle G$.

    $\displaystyle \forall a,b \in G$, define relation $\displaystyle R$ as $\displaystyle aRb$ if $\displaystyle aH=bH$.
    Relation $\displaystyle R$ is called a 'coset'.

    What does that relation, coset, mean? I interpret the definition as "$\displaystyle a$ relation $\displaystyle b$ if $\displaystyle a$ left times a subgroup equals $\displaystyle b$ left times the same subgroup". But that is kind of meaningless. What is an element times a subgroup?

    Thanks, in advance,
    Jeff
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  2. #2
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    Yes this is a coset, its purpose or application is probably not that important at this stage.

    I would make sure you understand how to find aH and Ha and the differences. (aH = Ha only when H is a normal subgroup of G)

    Do you have an example?
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  3. #3
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    No example. I typed everything in my notes (and I'm listening to my recording of the lecture now, there is nothing more in what the Prof. said).

    I guess my question was more along the lines of, what is a coset? What is $\displaystyle aH$? Is it just multiplying $\displaystyle a$ times $\displaystyle H$? If yes, what does it mean to multiply an element by a set? Does that simply mean multiplying $\displaystyle a$ times every element of the set? If yes, doesn't that imply that the coset is not a subset of the subgroup.

    The textbook does not have any information on cosets and I'm loathe to look at Wikipedia (it often confuses me worse).

    Thanks,
    P.S. What happens if I agree with you? (re: your signature)
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  4. #4
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    Quote Originally Posted by MSUMathStdnt View Post
    From my Discrete class notes:
    Let $\displaystyle G$ be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let $\displaystyle H$ be a subgroup of $\displaystyle G$.
    $\displaystyle \forall a,b \in G$, define relation $\displaystyle R$ as $\displaystyle aRb$ if $\displaystyle aH=bH$.
    Relation $\displaystyle R$ is called a 'left coset'.
    What does that relation, coset, mean?
    First of all understand what $\displaystyle aH$ means.
    $\displaystyle aH=\{ah:h\in H\}$ in other words $\displaystyle aH$ is simply a set obtained by operating $\displaystyle a$ on each element of $\displaystyle H$.
    That is called a left-coset of $\displaystyle H$ generated by $\displaystyle a$.

    Now we say that $\displaystyle aRb$ if and only if $\displaystyle aH=bH$.
    i.e. the generate the same left-coset.
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  5. #5
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    Quote Originally Posted by Plato View Post
    First of all understand what $\displaystyle aH$ means.
    $\displaystyle aH=\{ah:h\in H\]$ in other words $\displaystyle aH$ is simply a set obtained by operating $\displaystyle a$ on each element of $\displaystyle H$.
    That is called a left-coset of $\displaystyle H$ generated by $\displaystyle a$.

    Now we say that $\displaystyle aRb$ if and only if $\displaystyle aH=bH$.
    i.e. the generate the same left-coset.
    OK. I was starting to suspect that. Thanks for confirmation and explanation.

    P.S. Was this a typo? Is there an extra 0 in there?
    $\displaystyle aH=\{ah:h\in H\]$

    Should read: $\displaystyle aH=\{ah:h\in H\}$ right?
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  6. #6
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    Yes, see my edit.
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