Homework: find right coset of a group

[annoymage]

Homework Statement

Let G be a group, H is subgroup of G, and [G]=2
find all the right and left coset of H in G

Homework Equations

n/a

The Attempt at a Solution

(finding right coset)

so there exist 2 distinct right coset, but how to find the 2 right coset?

let a,b in G

so Ha and Hb are the right coset if Ha$$\cap$$Hb={}

then?

 

[Dick]

Pick an a in G that is not in H. Then the right cosets are H and Ha, right?

 

[annoymage]

yes, and is that the answer?

let a in G not in H

H and Ha are the right coset of H in G
H and aH are the left coset of H in G
is that correct and sufficient?

what if i do like this

let ab^-1 in G not in H

Ha and Hb are the right coset of H in G
aH and bH are the left coset of H in G
it's the same thing right?

 

[Dick]

Same thing, yes. But I don't know that it really helps you in any way. I think the point is that there is only one right coset that is not equal to H and there is only one left coset that is is not equal to H. So they must be equal. Isn't that the point?

 

[annoymage]

yea, haha, like you said, i wanted to show Hx=xH for all x in G, thank you very much,