Exercise 1.6.8 (If $n
e m$, $S_n$ and $S_m$ are not isomorphic)

Question

Prove that if $n 
e m$ then $S_n$ and $S_m$ are not isomorphic.

Richard Ganaye · Accepted Answer

\begin{proof} 
If $n < m$, then $m! = m(m-1)\cdots (m-n+1) n! >n!$, so the sequence $(n!)$ is strictly increasing, a fortiori injective. Therefore
$$n 
e m \Rightarrow n! 
e m! \Rightarrow |S_n| 
e |S_m|.$$
Since $S_n$ and $S_m$ have not the same cardinality, they are not isomorphic.
\end{proof}

Exercise 1.6.8 (If $n\ne m$, $S_n$ and $S_m$ are not isomorphic)

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Exercise 1.6.8 (If $n\ne m$, $S_n$ and $S_m$ are not isomorphic)

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