Centralizer of Subset is Intersection of Centralizers of Elements

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Theorem

Let $\left({G, \circ}\right)$ be a group.

Let $S \subseteq G$.


Let $C_G \left({S}\right)$ be the centralizer of $S$ in $G$.

Then:

$\displaystyle C_G \left({S}\right) = \bigcap_{x \mathop \in S} C_G \left({x}\right)$

where $C_G \left({x}\right)$ is the centralizer of $x$ in $G$.


Proof

By definition of the centralizer of $S$ in $G$:

$C_G \left({S}\right) = \left\{{g \in G: \forall s \in S: x \circ s = s \circ x}\right\}$



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