# Are these equivalent definitions of a quotient group?

After grappling with the concept of a quotient group I have come across two different definitions of a quotient group. The first which is used in Algebra Chapter 0 uses an equivalence relation $\sim$ on the underlying set of a group $G$ to obtain $G/{\sim}$ and then defines an operation on the equivalence classes. But the definition on Mathworld says a quotient group $G/N$ by a normal subgroup $N$ is simply the set of all cosets of $N$ in $G$ where the operation is coset multiplication , which doesn't even mention an equivalence relation. My question is are these definitions (in your opinion) equivalent and which do you feel is more useful in defining a quotient ring? Thanks

• Yes they are equivalent. It would be a good exercise for you to determine why---in particular, try to find an equivalent relation on $G$ whose equivalence classes are precisely the left cosets of $N$. Commented Mar 3, 2015 at 2:38

## 1 Answer

Yes, they are equivalent. The normal subgroup $N$ corresponds to the equivalence class of $e$ in the first variant, and the equivalence class version can be recovered from $N$ by using the equivalence relation:

$g_1 \sim_N g_2 \iff g_1^{-1}g_2 \in N$

EDIT: Personally I prefer to view a quotient group $G/N$ as the image $\phi(G)$ of a homomorphism:

$\phi: G \to G'$

The equivalence class view generalizes more nicely to monoids, whereas the "normal subgroup/kernel" view generalizes more nicely to rings and modules.

• Even better: a quotient group is a surjective homomorphism. This gives a nice symmetry, since a subgroup is an injective homomorphism. Commented Mar 3, 2015 at 3:04