Question

Prove that an element $\mathit{a}$is in the center of a group G if and only if $\mathit{C}\left(a\right)\mathbf{=}\mathit{G}$(notation as in Exercise 33).

Short answer

Itis proved that an element a is in the center of a group G if and only if $C\left(a\right)=G$.

Step-by-step solution

Centralizer and center of a group

Consider an element $a$of group G. Then, the centralizer of the element a is the set defined as.

$C\left(a\right)=\left\{g\in G|ga=ag\right\}$

The center of the group $G$is represented by term $Z\left(G\right)$and defined as a set of elements of $G$, which commute with every element of $G$.

Prove that an element is in the center of group G if and only if C(a)=G .

Assume that a is in the center of group G . Then, it will be true that ab=ba for all $b\in G$.

Hence,$C\left(a\right)=G$ because $C\left(a\right)$is a subset of$G$ , but all the elements from $G$commute with every element from $G$.

Assume that $C\left(a\right)=G$. Then, by definition, it is known that $ga=ag$for all$g\in G$ . Thus,$a$ is in the center of group $G$by definition.

Therefore, it is proved that an element $a$is in the center of group G if and only if $C\left(a\right)=G$.