The following theorem is taken from the german book Theorie der endlichen Gruppen by Kurzweil/Stellmacher. I am highly suspicious about a claim and its proof from the chapter 1.6 on products of groups, and just want to know if this is justified and if my alternate reasoning works.
First a direct translation of the theorem and its proof.
Let $G$ be the product of normal subgroups $G_1, \ldots, G_n$. Let $$ Z_i := G_i \cap \prod_{j\ne i} G_j, \quad i = 1, \ldots, n. $$ Then $$ G / \cap_{i=1}^n Z_i \cong G_1/Z_1 \times \ldots \times G_n/Z_n. $$ Proof. For $g \in G$ we have $g_i \in G_i$ with $$ g = g_1\cdots g_n. $$ Because the subgroups $G_1, \ldots, G_n$ are normal, for $i = 1,\ldots, n$ the mapping $$ \beta_i : G \to G/Z_i \quad \mbox{with}\quad g \mapsto g_i Z_i $$ is well-defined and a homomorphism with $\operatorname{Kern}(\beta_i) = Z_i$. Then $$ \alpha : G \to G/Z_1 \times \ldots G/Z_n \mbox{ with } g \mapsto (g_1Z_1, \ldots, g_nZ_n) $$ is a homomorphism with $$ \operatorname{Kern}(\alpha) = \bigcap_{i=1}^n \operatorname{Kern}(\beta_i) = \bigcap_{i=1}^n Z_i. $$ Hence the claim follows from the fundamental homomorphism theorem. $\square$
First I believe the correct claim should be $$ G / \prod_{i=1}^n Z_i \cong G_1/Z_1 \times \ldots \times G_n/Z_n. $$ After a close look at the proof, I am suspicious about the claim that the kernel of $\beta_i$ is $Z_i$. This would be the case for the projection map $g \mapsto gZ_i$, but here we pick out an element from some product decomposition, hence for example $g_1 \in G_1$ would also be mapped to $1\cdot Z_2$ by $\beta_2$ as $g_1 = g_1 \cdot 1$ with $1 \in G_2$. To be more specific I believe the kernel should be $$ \operatorname{Kern}(\beta_i) = Z_i \cup \prod_{j\ne i} G_j = \prod_{j\ne i} G_j. $$ Hence following the proof we would get $$ \operatorname{Kern}(\alpha) = \bigcap_{i=1}^n \left( \prod_{j\ne i} G_j \right) $$ Now if we fix $i \in \{1,\ldots, n\}$ then $$ Z_i \le \prod_{j \ne k} G_j $$ for any other $k \in \{1,\ldots, n\}$, hence $Z_1 \cdots Z_n \subseteq \operatorname{Kern}(\alpha)$. The converse implication is also valid, but this is a tedious chain of applications of Dedekinds law for groups, for example $Z_1 Z_2 Z_3 = (G_1 \cap G_2 G_3)(G_2 \cap G_1 G_3)(G_3\cap G_1G_2) = G_1G_2 \cap G_1G_3 \cap G_2 G_3$. Hence $\operatorname{Kern}(\alpha) = \prod_{i=1}^n Z_i$.
But there is also a more direct way to see this \begin{align*} g = g_1 \ldots g_n \in \operatorname{Kern}(\alpha) & \Leftrightarrow g_1 \in Z_1, \ldots, g_n \in Z_n \\ & \Leftrightarrow g \in Z_1 \ldots Z_n. \end{align*}
Seems to be quite a subtle point, and actually it came to my mind in later chapters related to components, the generalized Fitting subgroup and the layer of a finite group where this theorem and implications for central products where used. And some arguments in the book confused me a little bit, hence it would be kind if you could help me resolve my confusion. Is it the case that the theorem is wrong? And is my reasoning correct then?
