Exercise 1.6.7 ($D_8$ and $Q_8$ are not isomorphic)

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

Richard Ganaye · Accepted Answer

\begin{proof} 
Using the Cayley tables of Exercise 1.5.2, we note that $D_8$ has $5$ elements of order $2$, which are $r^2, s ,rs, r^2s, r^3s$. But $Q_8$ has only one element of order $2$, namely $-1$. By Exercise 2, $D_8$ and $Q_8$ are not isomorphic.
\end{proof}

Exercise 1.6.7 ($D_8$ and $Q_8$ are not isomorphic)

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Exercise 1.6.7 ($D_8$ and $Q_8$ are not isomorphic)

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