Let $N_1, . . . , N_n$ be normal subgroups of $G$, consider $G/N_1\times ··· \times G/N_n$

I’m reading Hans Kurzweil ‘s “The Theory of Finite Groups”, where it says

1.6.4 Let $ N_1, . . . , N_n$ be normal subgroups of $ G$ . Then the mapping $ $ α: G→G/N_1\times ··· \times G/N_n$ $ given by $ $ g \mapsto (gN_1,…,gN_n)$ $ is a homomorphism with $ Ker α = \cap_i N_i$ . In particular, $ G/\cap_i N_i$ is isomorphic to a subgroup of $ G/N_1 \times ··· \times G/N_n$ .

I’m confused here: can we write $ $ G/N_1\times ··· \times G/N_n$ $ ? To write a product of groups as this, it’s required that each $ G/N_i$ has only $ e$ as common element.

What if $ $ G=C_2 \times C_3 \times C_5 \times C_7$ $

$ $ N_1=C_2 \times C_3 $ $

$ $ N_2=C_2 \times C_5 $ $

$ $ N_3=C_2 \times C_7 $ $

, shouldn’t $ $ G/N_1 \congs C_3 \times C_5$ $

$ $ G/N_2 \congs C_2 \times C_7$ $

$ $ G/N_3 \congs C_5 \times C_7$ $

, and they have common elements besides $ e$ ?