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 |
$\blacksquare$
Sources
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms: Exercise $(11)$