Thread: 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 be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let be a subgroup of .

, define relation as if .
Relation is called a 'coset'.

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

Thanks, in advance,
Jeff

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?

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 ? Is it just multiplying times ? If yes, what does it mean to multiply an element by a set? Does that simply mean multiplying 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)

Originally Posted by MSUMathStdnt

From my Discrete class notes:
Let be a group (a set with an operation that is closed and associative, has an identity and every element has an inverse). Let be a subgroup of .
, define relation as if .
Relation is called a 'left coset'.
What does that relation, coset, mean?

First of all understand what means.
in other words is simply a set obtained by operating on each element of .
That is called a left-coset of generated by .

Now we say that if and only if .
i.e. the generate the same left-coset.

Originally Posted by Plato

First of all understand what means.
in other words is simply a set obtained by operating on each element of .
That is called a left-coset of generated by .

Now we say that if and only if .
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?


Should read: right?

Yes, see my edit.