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Math Help - 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 G be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let H be a subgroup of G.

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

    What does that relation, coset, mean? I interpret the definition as " a relation b if a left times a subgroup equals 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 aH? Is it just multiplying a times H? If yes, what does it mean to multiply an element by a set? Does that simply mean multiplying 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 G be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let H be a subgroup of G.
    \forall a,b \in G, define relation R as aRb if aH=bH.
    Relation R is called a 'left coset'.
    What does that relation, coset, mean?
    First of all understand what aH means.
    aH=\{ah:h\in H\} in other words aH is simply a set obtained by operating a on each element of H.
    That is called a left-coset of H generated by a.

    Now we say that aRb if and only if 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 aH means.
    aH=\{ah:h\in H\] in other words aH is simply a set obtained by operating a on each element of H.
    That is called a left-coset of H generated by a.

    Now we say that aRb if and only if 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?
    aH=\{ah:h\in H\]

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