Inner Automorphism Maps Subgroup to Itself iff Normal/Sufficient Condition
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Theorem
Let $G$ be a group.
For $x \in G$, let $\kappa_x$ denote the inner automorphism of $x$ in $G$.
Let $H$ be a normal subgroup of $G$.
Then:
- $\forall x \in G: \kappa_x \sqbrk H = H$
Proof
Let $H$ be a normal subgroup of $G$.
Let $x \in G$ be arbitrary.
By definition, $\kappa_x: G \to G$ is a mapping defined as:
- $\forall g \in G: \map {\kappa_x} g = x g x^{-1}$
Let $n \in N$.
Then:
\(\ds \map {\kappa_x} n\) | \(=\) | \(\ds x n x^{-1}\) | ||||||||||||
\(\ds \) | \(\in\) | \(\ds N\) | Definition of Normal Subgroup |
$\blacksquare$
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.10$: Theorem $27$