Understanding Notation G/H [SOLVED] set of cosets

[ehrenfest]

[SOLVED] set of cosets

Homework Statement

Does the notation G/H mean the set right cosets or the set of left cosets of H in G (where H is a subgroup of a group G)?

I've seen both definitions on the internet, but maybe I am just looking at bad sites.

Homework Equations

The Attempt at a Solution

 

[HallsofIvy]

Go back and read your textbook again. The only time the notation "G/H" makes sense (i.e. is a group itself) is when H is a normal subgroup of G. And that only happens when the left cosets and the right cosets are the same.

 

[ehrenfest]

The internet says otherwise:
http://web.ew.usna.edu/~wdj/book/node181.html

 

[Kreizhn]

I think then that it's just a notational difference. What the website show's though is quite confusing. From what I've seen, we traditionally denote the quotient group by G/H, and like HallsofIvy says, H has to be a normal subgroup of G, and in that case the left and right cosets are the same.

 

[Kreizhn]

I suppose that this is perhaps a generalization of the notation used for a quotient group, since the quotient group seems to be a special case of the notation when H is normal to G. (as the author stipulates in the first "notation" segment.)

 

[HallsofIvy]

Okay. As soon as I wrote that, I knew I was being too vague. Given a group G and subgroup H, you really should say "the set (better "collection") of left cosets or right cosets". But just about the only reason for distinguishing between the two is when you are trying to determine if the two are the same.

 

[ehrenfest]

Okay. As soon as I wrote that, I knew I was being too vague. Given a group G and subgroup H, you really should say "the set (better "collection") of left cosets or right cosets". But just about the only reason for distinguishing between the two is when you are trying to determine if the two are the same.

I am getting confused. It is true that the collection of left cosets and the collection of right cosets are NOT in general the same collection, agreed?

That means there might be theorems about the collection of left cosets that do not apply to the collection of right cosets, so there might be other reasons for distinguishing them, right?

 

[d_leet]

I am getting confused. It is true that the collection of left cosets and the collection of right cosets are NOT in general the same collection, agreed?

Yes, but the point to note is that if the subgroup H is normal in G then the two collections will be the same.

 

[ehrenfest]

Then I do not understand why people are saying that there is no reason to distinguish left and right cosets.

I also do not see why people are saying that G/H only makes sense when H is normal.

 

[d_leet]

Then I do not understand why people are saying that there is no reason to distinguish left and right cosets.

I also do not see why people are saying that G/H only makes sense when H is normal.

HallsofIvy pointed out that usually the only reason to distinguish between the two is to determine if they are the same.

And G/H only makes sense if H is normal because it is a group only if H is normal in G.

 

[Dick]

Then I do not understand why people are saying that there is no reason to distinguish left and right cosets.

I also do not see why people are saying that G/H only makes sense when H is normal.

There are a perfectly good reasons to distinguish left and right cosets. There is just no terribly standard notation for them.