Normal Subgroup of Subset Product of Subgroups

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Theorem

Let $G$ be a group whose identity is $e$.

Let:

$H$ be a subgroup of $G$
$N$ be a normal subgroup of $G$.


Then:

$N \lhd N H$

where:

$\lhd$ denotes normal subgroup
$N H$ denotes subset product.


Proof

From Subset Product with Normal Subgroup is Subgroup:

$N H = H N$ is a subgroup of $G$.

By definition of subset product all elements of $H N$ can be written in the form:

$h n \in H N$

where $h \in H, n \in N$.

Let $h n \in H N$.

Let $n_1 \in N$.


From Inverse of Group Product:

$\paren {h n} n_1 \paren {h n}^{-1} = h n n_1 n^{-1} h^{-1}$

We have that:

$n n_1 n \in N$
$h, h^{-1} \in G$.

Then, since $N$ is a normal subgroup of $G$:

$\paren {h n} n_1 \paren {n^{-1} h^{-1} } \in N$

Thus:

$N \lhd N H$

$\blacksquare$


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