Exhibit the cyclic subgroups of Dih(8) as sets

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 2.3 Exercise 2.3.11
Find all cyclic subgroups of $D_8$. Exhibit a proper subgroup of $D_8$ which is not cyclic.

Solution: We have the following. $$\langle 1 \rangle = \{ 1 \}$$ $$\langle r \rangle = \{ 1, r, r^2, r^3 \}$$ $$\langle r^2 \rangle = \{ 1, r^2 \}$$ $$\langle r^3 \rangle = \{ 1, r, r^2, r^3 \}$$ $$\langle s \rangle = \{ 1, s \}$$ $$\langle sr \rangle = \{ 1, sr \}$$ $$\langle sr^2 \rangle = \{ 1, sr^2 \}$$ $$\langle sr^3 \rangle = \{ 1, sr^3 \}$$We saw in a previous exercise that $\{ 1, r^2, s, r^2s \}$ is a subgroup of $D_8$, but is not on the above list, hence is not cyclic.