understanding the language of a theorem The theorem is stated as follows in the book:

Let $\phi:G\rightarrow G'$ be a group homomorphism, and let
  $H=Ker(\phi)$. Let $a\in G$. Then the set 
$\phi^{-1}[\{\phi(a)\}] = \{x\in G | \phi(x)=\phi(a)\}$
is the left coset $aH$ of $H$, and is also the right coset $Ha$ of
  $H$. Consequently, the two partitions of $G$ into left cosets and into
  right cosets of $H$ are the same.

I'm trying to parse this statement and it's not clear to me what claim the author is trying to make at the very end when he says "Consequently, the two partitions of $G$ into left cosets and into right cosets of $H$ are the same." I'm under the impression that, in general, the left and right cosets are not always the same. Under what condition are they the same? Under the condition that you have a homomorphism? 
Let me mention that at this point, we're not supposed to know what a normal subgroup is. The author introduces the idea of a normal subgroup 2 pages later.

What the statement is saying is that the kernel of a homomorphism is a normal subgroup. 
I don't know if this clarifies or not, as I don't know the context: it could be that the author is trying to use this idea to introduce the notion of normal subgroup. 
