A7 - center

center

the center, $Z (G)$ , of a group $G$ is the subset of elements in $G$ that commute with every element of $G$

Z (G) = {a \in G ∣ a x = x a \forall z \in G}

center is a subgroup

the center of a group $G$ is a subgroup of $G$

Z (G) \leq G

centralizer

let $a \in G$ , then its centralizer $C (a)$ is the set of all elements in $G$ that commute with $a$

C (a) = {g \in G ∣ g a = a g}

C (a)

is a subgroup

for each $a \in G$ , the centralizer of $a$ is a subgroup of $G$

for every element $a \in G$ , $Z (G) \subseteq C (a)$
$G$ is Abelian iff $C (a) = G \forall a \in G$