Centralizer is Normal Subgroup of Normalizer

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Theorem

Let $G$ be a group.

Let $H \le G$ be a subgroup of $G$.

Let $\map {C_G} H$ be the centralizer of $H$ in $G$.

Let $\map {N_G} H$ be the normalizer of $H$ in $G$.

Let $\Aut H$ be the automorphism group of $H$.


Then:

$(1): \quad \map {C_G} H \lhd \map {N_G} H$
$(2): \quad \map {N_G} H / \map {C_G} H \cong K$

where:

$\map {N_G} H / \map {C_G} H$ is the quotient group of $\map {N_G} H$ by $\map {C_G} H$
$K$ is a subgroup of $\Aut H$.


Proof

In order to invoke the First Isomorphism Theorem for Groups, we must construct a group homomorphism $\phi: \map {N_G} H \to \Aut H$.

Consider the mapping $\phi: x \mapsto \paren {g \mapsto x g x^{-1}}$.

From Inner Automorphism is Automorphism, $g \mapsto x g x^{-1}$ is an automorphism of $G$, so $\phi$ is well-defined.


To see that $\phi$ is a homomorphism, notice that for any $x, y \in \map {N_G} H$:

\(\ds \map \phi x \map \phi y\) \(=\) \(\ds \paren {g \mapsto x g x^{-1} } \circ \paren {g \mapsto y g y^{-1} }\) where $\circ$ denote composition of maps
\(\ds \) \(=\) \(\ds g \mapsto x \paren {y g y^{-1} } x^{-1}\)
\(\ds \) \(=\) \(\ds g \mapsto \paren {x y} g \paren {x y}^{-1}\) Inverse of Group Product
\(\ds \) \(=\) \(\ds \map \phi {x y}\)

Hence $\phi$ is a homomorphism.


Now we prove that $\ker \phi = \map {C_G} H$.

Note that for $x \in \map {N_G} H$:

\(\ds x\) \(\in\) \(\ds \ker \phi\)
\(\ds \leadstoandfrom \ \ \) \(\ds g\) \(=\) \(\ds x g x^{-1}\) \(\ds \forall g \in H\) $g \mapsto g$ is the identity of $\Aut H$
\(\ds \leadstoandfrom \ \ \) \(\ds g x\) \(=\) \(\ds x g\) \(\ds \forall g \in H\)
\(\ds \leadstoandfrom \ \ \) \(\ds x\) \(\in\) \(\ds \map {C_G} H\) Definition of Centralizer of Subgroup

Hence $\ker \phi = \map {C_G} H$.


By Kernel is Normal Subgroup of Domain:

$\map {C_G} H \lhd \map {N_G} H$

By First Isomorphism Theorem for Groups:

$\map {N_G} H / \map {C_G} H \cong \Img \phi$

By Image of Group Homomorphism is Subgroup:

$\Img \phi \le \Aut H$

Hence the result.

$\blacksquare$


Sources

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