Let $G$ be a D22: Group such that

(i) | $Z(G)$ is the D1563: Group centre of $G$ |

Then $Z(G)$ is an D23: Abelian group.

Result R755
on D1563: Group centre

Group centre is Abelian group

Formulation 0

Let $G$ be a D22: Group such that

(i) | $Z(G)$ is the D1563: Group centre of $G$ |

Then $Z(G)$ is an D23: Abelian group.

Proofs

Let $G$ be a D22: Group such that

(i) | $Z(G)$ is the D1563: Group centre of $G$ |

First of all, result R810: Group centre is nonempty shows that $Z(G)$ is not empty. Therefore, if $g \in Z(G)$, then $g x g^{-1} = x$ and thus $g x = x g$ for every $x \in G$. Particularly, $g x = x g$ for every $x \in Z(G)$, which proves the claim. $\square$