Let $G$ be a finite group. We will say that $G=A \times B \times C$ if
- A,B,C are normal in $G$
- $A\cap B \cap C ={e}$
- $|G|=|A||B||C|$
Is the first condition ok? or should I say $A \times B$ is normal in $G$ and $C$ is normal in $G$.
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That is satisfied if $G$ is an elementary Abelian group of order $8$, $A=\{e,a\}$, $B=\{e,b\}$ and $C=\{e,ab\}$ where $a$ and $b$ are distinct elements of order two. But in this case the natural homomorphism from $A\times B\times C$ to $G$ is not an isomorphism.
In this case the subgroups "$A\times B$", "$A\times C$" and "$B\times C$" are all normal in $G$ too.
Angina Seng
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I did not get your answer. In general, Is first condition fine? – I_wil_break_you as Jan 24 '19 at 06:25
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In an Abelian group all subgroups are normal. @I_wil_break_wall – Angina Seng Jan 24 '19 at 06:26
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My question is what are the conditions for $G$ can be written as a direct product of three subgroups. – I_wil_break_you as Jan 24 '19 at 06:27