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The centralizer $C_{G} (H)$ of a subgroup

Definition

Let $G$ be a group (Group and abelian group) and let $H \leq G$ . The centralizer of $H$ in $G$ is

$C_{G} (H) : = {g \in G : g h = h g for every h \in H} .$

Equivalently, $C_{G} (H) = ⋂_{h \in H} C_{G} (h)$ , the intersection of the centralizers of the individual elements of $H$ (The conjugacy class ${Cl}_{G} (x)$ and centralizer $C_{G} (x)$ of an element).

Why it is a subgroup. The identity satisfies $e h = h = h e$ for every $h \in H$ , so $e \in C_{G} (H)$ . If $g_{1}, g_{2} \in C_{G} (H)$ and $h \in H$ , then

$(g_{1} g_{2}) h = g_{1} (g_{2} h) = g_{1} (h g_{2}) = (g_{1} h) g_{2} = (h g_{1}) g_{2} = h (g_{1} g_{2}),$

so $g_{1} g_{2} \in C_{G} (H)$ . If $g \in C_{G} (H)$ and $h \in H$ , then multiplying $g h = h g$ by $g^{- 1}$ on both sides gives $h g^{- 1} = g^{- 1} h$ , so $g^{- 1} \in C_{G} (H)$ . Hence $C_{G} (H) \leq G$ .

Two special cases are used without further comment. Taking $H = G$ gives $C_{G} (G) = Z (G)$ , the center (The center $Z (G)$ of a group). Intersecting with $H$ gives $C_{G} (H) \cap H = Z (H)$ , since an element of $H$ lies in $C_{G} (H)$ exactly when it commutes with every element of $H$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. S. Milne, Group Theory, Proposition 3.22 (standard reference, not scraped)

The centralizer CG(H)CG​(H) of a subgroup