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Demonstrate that given subsets of Dih(8) are subgroups


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.

Linearity

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