Centralizer of Subset is Intersection of Centralizers of Elements

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Theorem

Let $\struct {G, \circ}$ be a group.

Let $S \subseteq G$.


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

Then:

$\ds \map {C_G} S = \bigcap_{x \mathop \in S} \map {C_G} x$

where $\map {C_G} z$ is the centralizer of $x$ in $G$.


Proof

\(\ds \map {C_G} S\) \(=\) \(\ds \set {g \in G: \forall x \in S: g \circ x = x \circ g}\) Definition of Centralizer of Group Subset
\(\ds \) \(=\) \(\ds \set {g \in G: \forall x \in S: g \in \map {C_G} x}\) Definition of Centralizer of Group Element
\(\ds \) \(=\) \(\ds \bigcap_{x \mathop \in S} \map {C_G} x\) Definition of Intersection of Family

$\blacksquare$


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