How do cosets form a group

Question

Let G be a group and H be a Subgroup . Then if H is normal then the cosets form a group . I cannot understand what is meant by cosets form a group . To from a group , group operation on two elements must lie inside the group.So when we say that cosets form a group , group operation on any two elements from any two cosets must lie inside the two cosets chosen earlier or the set of cosets (which is group G itself) ?

Answer 1

To from a group , group operation on two elements must lie inside the group.

Yes, it means that $\{aH\mid a\in G \}$ has an operation that makes it a group, namely $aH\cdot bH=abH$. This operation does not work unless $H$ is normal.

The way to think of it is that the cosets are "big points" which are globbed together from little points. They are "new points" in a different group, and their multiplication is related to the multiplication in the original group by the simple rule above.

Now, perhaps you're feeling a little disoriented because the new "points" are themselves sets of points. This is a common feeling at the beginning, when one has for so long thought of points and sets of points and nothing "stacked deeper" than that. But if you study a bit of basic set theory, you'll find that almost everything consists of variously stacked sets, and that it's not so unusual. Try to get comfortable of this idea of "big points" and simply forgetting what their internals used to be.

So when we say that cosets form a group , group operation on any two elements from any two cosets must lie inside the two cosets chosen earlier or the set of cosets (which is group G itself) ?

It turns out that yes, this "setwise" multiplication of elements from $aH$ and $bH$ equals $abH$, but in my opinion, this is never useful. In fact, it is counterproductive when one moves on to quotients of rings, where the intuition applied to multiplication in the quotient ring is simply wrong.