## 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 $$e$$?