Intersection of cosets is empty or a coset

[Dragonfall]

"Let H and K be subgroups of a group G. Prove that the intersection $$xH\cap yK$$ of two cosets of H and K is either empty or is a coset of the subgroup $$H\cap K$$."

I'm stuck here.

 

[matt grime]

How are you stuck? You're supposed to say what you've done, in order to get help. So, try it. Suppose it is not empty, what can you show? What do you need to show? Where do you get stuck in going from what you know to what you need to show? Have you written down the definition of what t

xHnyK

and a coset of

HnK

are? Because I think if you do the line of argument becomes clear.

 

[Dragonfall]

Suppose it's not empty, then there exists h in H and k in K such that xh=yk. I need to show that every such element can be written as za=xh=yk for some z in G and some a in HnK.