Problem 3

Let $G$ be a group.

1. Prove that $\mathrm{Z}(G)$ is a normal subgroup of $G$.

2. Prove that $\mathrm{Z}(G)=G$ if and only if $G$ is abelian.

3. Find the center of the group $D_{4}$.(Note: you may assume $|\mathrm{Z}(G)|<\frac{1}{2}|G|$. The reason becomes apparent with 1.11’s solution.)