Direct Product of Normal Subgroups is Normal

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Theorem

Let $G$ and $G'$ be groups.

Let:

$H \lhd G$
$H' \lhd G'$

where $\lhd$ denotes the relation of being a normal subgroup.


Then:

$\paren {H \times H'} \lhd \paren {G \times G'}$

where $H \times H'$ denotes the group direct product of $H$ and $H'$


Proof

Let $\tuple {x, x'} \in G \times G'$ and $\tuple {y, y'} \in H \times H'$.

Then:

\(\ds \tuple {x, x'} \tuple {y, y'} \tuple {x, x'}^{-1}\) \(=\) \(\ds \tuple {x, x'} \tuple {y, y'} \tuple {x^{-1}, x'^{-1} }\)
\(\ds \) \(=\) \(\ds \tuple {x y x^{-1}, x' y' x'^{-1} }\)
\(\ds \) \(\in\) \(\ds H \times H'\) Definition of Normal Subgroup

Hence:

$\paren {H \times H'} \lhd {G \times G'}$

$\blacksquare$


Sources

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