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## Dih(8) and the quaternion group are not isomorphic

Prove that $D_8$ and $Q_8$ are not isomorphic.

Solution: We saw in Exercise 1.5.2 that $D_8$ has five elements of order 2, namely, $sr^k$ for $0 \leq k < 4$ and $r^2$, but that $Q_8$ has only one element of order 2, namely $-1$.