**Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.2 Exercise 2.2.7**

Let $n \in \mathbb{Z}$ with $n \geq 3$. Prove the following.

(1) $Z(D_{2n}) = 1$ if $n$ is odd.

(2) $Z(D_{2n}) = \{ 1, r^k \}$ if $n = 2k$ is even.

Solution:

(1) This is the content of Exercise 1.2.4.

(2) This is the content of Exercise 1.2.5.