Center of Group is Abelian Subgroup

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $G$ be a group.

Let $\map Z G$ be the center of $G$.

Then $\map Z G$ is an abelian subgroup of $G$.


Proof

By Center of Group is Subgroup, $\map Z G$ is a subgroup of $G$.


The definition of the center $\map Z G$ grants that all elements of $\map Z G)$ commute with all elements of $G$.

In particular, all elements of $\map Z G$ commute with all elements of $\map Z G$ as $\map Z G \subseteq G$.

Therefore $\map Z G$ is abelian.

$\blacksquare$


Sources