# Thread: What does this relation mean?

1. ## 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?

Jeff

2. 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?

3. 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)

4. Originally Posted by MSUMathStdnt
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.

5. Originally Posted by Plato
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?

6. Yes, see my edit.