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

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