Computing modulo a normal subgroup behaves well, as becomes clear by the following result.
Proposition
Let N be a normal subgroup of the group G.
The map p : G -> G/N, g -> gN is a surjective morphism with kernel N.
Let f : G -> H be a surjective morphism between the two groups G and H with kernel N. According to a previous proposition, N is a normal subgroup of G.
First isomorphism theorem for
groups
If G and H are groups and f : G -> H is a surjective morphism with kernel N, then the map f': G/N -> H defined by f'(gN) = f(g) is an isomorphism.
According to the theorem, computing in H is essentially
the same as computing in G modulo N.
