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Let $G$ be a group (finite ) and let $G$ be a center of $Z$, now assume that $G = G_1 \times G_2 \cdots G_k$ is the decomposition of $G$, where each $G_i$ is indecomposable.

Is it true that

$G /Z = G_1/Z \times G_2/Z \times ...G_k/Z$ ?

or just tell the correct relation here.

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closed as off-topic by Saad, JMP, Dietrich Burde, Derek Holt, Carl Mummert Mar 14 '18 at 18:43

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It is clear that $Z(G)=Z(G_1)\times Z(G_2)\times ...\times Z(G_k)$. Consider the map $\phi : G\rightarrow G_1/Z(G_1)\times ...\times G_k/Z(G_k)$ with $\phi(g)=(\overline{g_1},...,\overline{g_k})$ where $g=(g_1,...,g_k)$. What is the kernel?

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  • $\begingroup$ please add more detail to your answer. I am naive in this field. $\endgroup$ – old Mar 14 '18 at 14:07
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    $\begingroup$ Which part is problematic? $\endgroup$ – Levent Mar 14 '18 at 14:08

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