Show that the following subsets of $D_8$ are subgroups.
(1) $\{ 1, r^2, s, sr^2 \}$
(2) $\{ 1, r^2, sr, sr^3 \}$.
Solution: Exercise 1.5.2 would be helpful!
(1) We have $r^2 r^2 = 1$, $r^2 s = sr^2$, $r^2 sr^2 = s$, $sr^2 = sr^2$, $ss = 1$, $ssr^2 = r^2$, $sr^2 r^2 = s$, $sr^2 s = r^2$, and $sr^2 sr^2 = 1$, so that this set is closed under multiplication. Moreover, $(r^2)^{-1} = r^2$, $s^{-1} = s$, and $(sr^2)^{-1} = sr^2$, so this set is closed under inversion. Thus it is a subgroup.
(2) We have $r^2 r^2 = 1$, $r^2 sr = sr^3$, $r^2 sr^3 = sr$, $srr^2 = sr^3$, $srsr = 1$, $sr sr^3 = r^2$, $sr^3 r^2 = sr$, $sr^3 sr = r^2$, and $sr^3 sr^3 = 1$, so that this set is closed under multiplication. Moreover, $(r^2)^{-1} = r^2$, $(sr)^{-1} = sr$, and $(sr^3)^{-1} = sr^3$, so this set is closed under inversion. Thus it is a subgroup.
