Lagrange's Theorem Group theory proof

Question

I am reading DF's proof of Lagrange's theorem that the order of a subgroup divides the order of a group.

The part where I cannot follow is when they say that :

The set of left cosets of H in G partition G (I can see this).

Bythe definition of a left coset the map:

$H \rightarrow gH$ defined by $h \mapsto gh$ is a surjection from H to the left coset gH (this I cannot see clearly) but I know the definition of a left coset is that you take all elements of G and multiply on the subgroup H on the left side

Injective since $gh_{1}=gh_{2} \implies h_{1}=h_{2}$

Then they conclude that H and gH have the same order I guess this is because we can see now it is bijective?

My question here is about the Surjectiveness, how can I see that? and my other question is what excatly are they doing when showing injectiveness are they just looking on the element map? \mapsto

Answer 1

Having looked at your post and flipped to the proof from Dummit and Foote that you're talking about, I have a hunch that you're getting hung up on the notation. That being said, perhaps it will help to rephrase the major points.

For any element $g \in G$, we can define the left-coset map $$ \phi_g : H \to gH\\ \phi_g(h) = gh $$ The image of this map is $$ \phi_g(H) = \{\phi_g(h) : h \in H\} = \{gh: h \in H\} $$ so that, by the definition of a left coset, $\phi_g(H) = gH$. In other words, the map $\phi_g$ is surjective.

Now, try to use the left-cancellation law to show that $\phi_g$ is injective.

Answer 2

I think misunderstanding is in statement "the definition of a left coset is that you take all elements of G".

No, definition if coset is that you take one element of G and multiply it by all element of $G$ (although different elements $g_1$ and $g_2$ can generate same coset). By taking all element of $G$ you create all cosets and $H$ itself (if $g \in H$).

So, function $H \rightarrow gH$ is surjective because for every $h$ exist element in $gH$ (it is $gh$), and as was mentioned by you injective