Exercise 1.6.9 ($D_{24}$ and $S_4$ are not isomorphic.)

Question

Prove that $D_{24}$ and $S_4$ are not isomorphic.

Richard Ganaye · Accepted Answer

\begin{proof} 
Note that $D_{24} = \langle r,s \mid r^{12} = s^2 = e,\ sr = r^{-1}s angle$ has an element of order $12$, because $r^{12} = e, r^3 
e e, r^4 
e e$.

But $S_4$ has no element of order $12$. By Exercise 1.3.4, the orders of the elements in $S_4$ are $1,2,3$ or $4$.

So by Exercise 2,  $D_{24} 
ot \simeq S_4$.
\end{proof}

Exercise 1.6.9 ($D_{24}$ and $S_4$ are not isomorphic.)

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Exercise 1.6.9 ($D_{24}$ and $S_4$ are not isomorphic.)

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