Exercise 1.3.8 ($S_{\mathbb{N}^*}$ is an infinite group)

Question

Prove that if $\Omega = \{1,2,3,\ldots\}$ then $S_{\Omega}$ is an infinite group (do not say $\infty!  =\infty$).

Richard Ganaye · Accepted Answer

\begin{proof} 
Let $S$ be the set of transpositions $	au_n = (1\ n)$ for all $n \geq 2$:
$$S =\{ (1\ 2), (1\ 3), (1\ 4),\ldots\} = \{	au_2,	au_3, 	au_4,\ldots\}.$$
These transpositions are distinct: if $	au_i =	au_j$ then $i = 	au_i(1) = 	au_j(1) = j$.

Therefore $S$ is an infinite set, and $S \subset \Omega$, so $S_{\Omega}$ is an infinite group.

\end{proof}

Exercise 1.3.8 ($S_{\mathbb{N}^*}$ is an infinite group)

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Exercise 1.3.8 ($S_{\mathbb{N}^*}$ is an infinite group)

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