02-02. Centralizers and normalizers, stabilizers and kernels

Definition. centlizer

• $G$,
  • group
• $A \neq \emptyset$,
  • subset

\[C_{G}(A) := \{ g \in G \mid \forall a \in A, \ gag^{-1} = a \}.\]

$C_{G}(A)$ is calledthe centralizer of $A$ in $G$.

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Proposition.

• $G$,
• $A \neq \emptyset \subseteq G$,
  • subset

\[g a g^{-1} = a \Leftrightarrow a g^{-1} = g^{-1} a \Leftrightarrow ga = ag .\]

proof.

$\Box$

Proposition.

• $G$,
• $A \neq \emptyset \subseteq G$,
  • subset

$C_{A}(G)$ is a subgroup of $G$.

proof.

$C_{A}(G) \neq \emptyset$ since $1 \in C_{G}(A)$. Suppose that $a, b \in C_{G}(A)$.

\[\begin{eqnarray} \forall c \in A, \ ab c (ab)^{-1} & = & a b c b^{-1} a^{-1} \nonumber \\ & = & a c a^{-1} \nonumber \\ & = & c \nonumber \end{eqnarray}\]

$C_{G}(A)$ is closed under the operation. Suppose that $a \in C_{G}(A)$.

\[\begin{eqnarray} \forall c \in A, \ a^{-1} c (a^{-1})^{-1} & = & a^{-1} c a \nonumber \\ & = & a^{-1} a c \quad (\because a \in C_{G}(A)) \nonumber \\ & = & c . \nonumber \end{eqnarray}\]

$\Box$

Definition. center

• $G$,
  • group

\[Z(G) := \{ g \in G \mid \forall g \in G, \ gx = xg \}\]

$Z(G)$ is called the center of $G$.

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Remark

\(C_{G}(G) = Z(G) .\)

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Definition. normalizer

• $G$,
  • group

\[\begin{eqnarray} gAg^{-1} & := & \{ gag^{-1} \mid a \in A \} \nonumber \\ N_{G}(A) & := & \{ g \in G \mid gAg^{-1} = A \} \nonumber \\ \end{eqnarray} .\]

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Remark

\(C_{G}(A) \subseteq N_{G}(A) .\)

$gag^{-1} \in A$ is weaker condition than $gag^{-1} = a$. $C_{G}(A)$ requires that $gag^{-1}$ exactly is equal to $a$.

\[\forall b \in B, \ bBb^{-1} \subseteq B \Rightarrow B \subseteq N_{G}(B) .\]

If $A \trianglelefteq G$, $N_{G}(A) = G$.

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