Definition:Centralizer

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

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

The centralizer of $$a$$ (in $$G$$) is defined as:

$$C_G \left({a}\right) = \left\{{x \in G: x \circ a = a \circ x}\right\}$$

That is, the centralizer of $$a$$ is the set of elements of $$G$$ which commute with $$a$$.

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

Let $$H \le \left({G, \circ}\right)$$.

The centralizer of $$H$$ (in $$G$$) is the set of elements of $$G$$ which commute with all $$h \in H$$:

$$C_G \left({H}\right) = \left\{{g \in G: \forall h \in H: g \circ h = h \circ g}\right\}$$

Centralizer of a Ring Subset
Let $$S$$ be a subset of a ring $$\left({R, +, \circ}\right)$$.

The centralizer of $$S$$ in $$R$$ is defined as:

$$C_R \left({S}\right) = \left\{{x \in R: \forall s \in S: s \circ x = x \circ s}\right\}$$

That is, the centralizer of $$S$$ is the set of elements of $$R$$ which commute with all elements of $$S$$.