# Direct Product of Normal Subgroups is Normal

## 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:

 $\displaystyle \tuple {x, x'} \tuple {y, y'} \tuple {x, x'}^{-1}$ $=$ $\displaystyle \tuple {x, x'} \tuple {y, y'} \tuple {x^{-1}, x'^{-1} }$ $\displaystyle$ $=$ $\displaystyle \tuple {x y x^{-1}, x' y' x'^{-1} }$ $\displaystyle$ $\in$ $\displaystyle H \times H'$ Definition of Normal Subgroup

Hence:

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

$\blacksquare$