Inner Automorphism Group is Isomorphic to Quotient Group with Center
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Theorem
Let $G$ be a group.
Let $\Inn G$ be the inner automorphism group of $G$.
Let $\map Z G$ be the center of $G$.
Let $G / \map Z G$ be the quotient group of $G$ by $\map Z G$.
Then $G / \map Z G \cong \Inn G$.
Proof
Let $G$ be a group.
Let the mapping $\kappa: G \to \Inn G$ be defined as:
- $\map \kappa a = \kappa_a$
where $\kappa_a$ is the inner automorphism of $G$ given by $a$.
From Kernel of Inner Automorphism Group is Center:
- $\map \ker \kappa = \map Z G$
and also that:
- $\Img \kappa = \Inn G$
From the First Isomorphism Theorem:
- $\Img \kappa \cong G / \map \ker \kappa$
Thus, as $\map \ker \kappa = \map Z G$ and $\Img \kappa = \Inn G$:
- $G / \map Z G \cong \Inn G$
$\blacksquare$
Sources
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $7$: Homomorphisms: Exercise $10$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 64 \beta$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $8$: Homomorphisms, Normal Subgroups and Quotient Groups: Exercise $25$