Centralizer

centralizer

For any subset $A \subseteq G$ , define $C_{G} (A) = {g \in G | g a g^{- 1} = a \forall a \in A}$ . This subset of $G$ is called the centralizer of $A$ in $G$ . Since $g a g^{- 1} = a$ iff $g a = a g$ , then $C_{G} (A)$ is the set of elements of $G$ which commute with every element in $A$ .

Note that $Z (G) = C_{G} (G)$ .

Intuition

The idea is that this is all elements such that they commute with all elements in $A$ . In contrast to the Normalizer, this means that:

g s = s g

if $g \in C_{G} (S)$ for some set $S \subseteq G$ .

To contrast this with the Normalizer, that will say that if $g \in N_{G} (S)$ then:

g s = t g

for some $t \in S$ instead. Namely, the element in $S$ can change as you apply commutativity. This is why we look at the Stabilizers too, since we could define centralizers in terms of a combination of normalizers and stabilizers.