1. **Problem statement:** Given that $n$ is odd, determine how many subgroups of the dihedral group $D_n$ are normal. 2. **Background:** The dihedral group $D_n$ is the group of symmetries of a regular $n$-gon, including $n$ rotations and $n$ reflections, so $|D_n|=2n$. 3. **Key fact:** When $n$ is odd, the only normal subgroups of $D_n$ are: - The trivial subgroup $\{e\}$, - The cyclic subgroup of rotations $\langle r \rangle$ of order $n$, - The whole group $D_n$ itself. 4. **Explanation:** - The subgroup of rotations $\langle r \rangle$ is normal because it is of index 2 in $D_n$. - No reflection-generated subgroups are normal when $n$ is odd. - The trivial subgroup and the whole group are always normal. 5. **Counting normal subgroups:** - $\{e\}$ (1 subgroup) - $\langle r \rangle$ (1 subgroup) - $D_n$ (1 subgroup) Total normal subgroups = 3. 6. **Check options:** - (a) $n$ (not necessarily 3) - (b) $\frac{n+1}{2}$ (not necessarily 3) - (c) 2 (incorrect, we have 3) - (d) 1 (incorrect) - None of the choices (correct) **Final answer:** None of the choices