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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 $\operatorname{Ker}α = \cap_i N_i$. 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 $$G/N_1\times \cdots \times G/N_n$$ ? 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 $$G/N_1 \cong C_3 \times C_5$$

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

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

, and they have common elements besides $e$?

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  • $\begingroup$ I believe this is the Chinese Reminder theorem for groups. $\endgroup$
    – IAmNoOne
    Jan 9, 2019 at 13:21
  • $\begingroup$ @IAmNoOne but there’s no solution formula? $\endgroup$
    – athos
    Jan 9, 2019 at 23:55

2 Answers 2

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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.

Note that $G/N_i$ is not a subgroup of $G $. So here we are not considering the internal direct product, which requires the condition you mentioned above to be a group. Here $G/N_1\times ··· \times G/N_n$ represents the external direct product , which is a group under the componentwise operation.

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Given two groups $G_1$ and $G_2$, you can form their direct product: $G_1×G_2$, to be $\{(g_1,g_2)\mid g_1\in G_1,g_2\in G_2\}$, with the group operation defined as $(g_1,g_2)+(h_1,h_2)=(g_1+h_1,g_2+h_2)$.

As an example, consider the Klein four group, $V_4=C_2×C_2$.

(As in the other answer, you seem to be thinking of the internal direct product of subgroups of the given group.)

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