Subset Products of Normal Subgroup with Normal Subgroup of Subgroup
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Theorem
Let $G$ be a group.
Let:
- $(1): \quad H$ be a subgroup of $G$
- $(2): \quad K$ be a normal subgroup of $H$
- $(3): \quad N$ be a normal subgroup of $G$
Then:
- $N K \lhd N H$
where:
- $N K$ and $N H$ denote subset product
- $\lhd$ denotes the relation of being a normal subgroup.
Proof
Consider arbitrary $x_n \in N, x_h \in H$.
Thus:
- $x_n x_h \in N H$
We aim to show that:
- $x_n x_h N K \paren {x_n x_h}^{-1} \subseteq N K$
thus demonstrating $N K \lhd N H$ by the Normal Subgroup Test.
We have:
\(\ds x_n x_h N K \paren {x_n x_h}^{-1}\) | \(=\) | \(\ds x_n x_h N K {x_h}^{-1} {x_n}^{-1}\) | Inverse of Group Product | |||||||||||
\(\ds \) | \(=\) | \(\ds x_n N x_h K {x_h}^{-1} {x_n}^{-1}\) | as $N \lhd G$ and $H \le G$ | |||||||||||
\(\ds \) | \(=\) | \(\ds x_n N K {x_n}^{-1}\) | as $K \lhd H$ | |||||||||||
\(\ds \) | \(=\) | \(\ds N K {x_n}^{-1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds K N {x_n}^{-1}\) | as $N \lhd G$ and $K \le G$ | |||||||||||
\(\ds \) | \(=\) | \(\ds K N\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds N K\) | as $N \lhd G$ and $K \le G$ |
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46 \eta$