I need suggestions for this exercise.
Let $D_{2n}$ stand for the dihedral group of $2n$ elements.
- Show that if $n$ is odd and $\forall b \in D_{2n} (ab = ba)$, then $a = e$.
- Show that if $n$ is even, then there is $a\in D_{2n}$ such that $\forall b \in D_{2n} (ab = ba)$ and $a \neq e$.
My ideas: The elements in $G$ can separated into two disjoint sets: $Q=\{a\in G : a^2\neq e\}$ and $G\setminus Q$. If $a\in Q$, then $a\neq a^{-1}$ and this implies that $a$ is not of order 2.
Also, any suggestions of introductory books on abstract algebra are welcome.
