# Formal definition of subgroup / quotient group duality

Recently I read that a quotient group is the dual of a subgroup. This resource describes a lot of their properties.

Wondering if:

• One could formally define both a subgroup and a quotient group without reference to each other (so quotient group is defined without subgroup). I am familiar with their basic definitions like on wikipedia.
• In a way that shows their duality (so the definitions are similar, as chain and cochain complex definitions are similar). If not possible, then how to get an intuition / formal definition of their duality.

The reason is a subgroup $N$ is defined simply as a subset where the group operation still works. But a quotient group is defined as all left cosets of a subgroup, i.e. $G/N = \{ aN : a \in G \}$. Not seeing the duality.

• The duality comes from the fact that essentially a subgroup is an injective group morphism $N\to G$, while a quotient group is a surjective group morphism $G\to K$. It remains to understand how "injective" is dual to "surjective" but that just stems from the fact that in the category of groups, monomorphisms are injective morphisms, and epimorphisms are surjective morphisms: epimorphisms and monomorphisms are clearly dual (see the definition); hence subgroups and quotient groups are dual – Max May 5 '18 at 9:26