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Suppose

$$G={\rm Sym}_{{\rm fin}}(\Bbb{N}):=\{\sigma\in S_\Bbb N\mid \sigma(x)=x \text{ for all but finitely many } x\in \Bbb{N}\}.$$

Prove $G$ is not finitely generated.

My solution:

Suppose $G$ is finitely generated, hence $G=\langle S\rangle$ such that $S\subset G$.

Since $S$ is a finite subset and there are $\infty$ numbers in $\mathbb{N}$, there is $x_1\in \mathbb{N}$ such that $\sigma(x_1)=x_1 \forall \sigma\in S$, $\sigma$ is a permutation of $\mathbb{N} .$

Obviously $\forall \sigma\in \langle S\rangle, \sigma(x_1)=x_1.$

On other side, denote $x_2\in \mathbb{N}$.

$\sigma'$ is a permutation of $\mathbb{N}$ $$\sigma'= \begin{cases} x_2,& x=x_1;\\ x_1,& x=x_2;\\ x,& x\neq x_1,x_2;\\ \end{cases}$$

Obviously, $\sigma'\in G$, contradiction.

Hence, $G$ is not finitely generated.

Is my solution correct?

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    $\begingroup$ Common questions may arise like "what is $\sigma$ here? (obvious guess is that $\sigma$ is a permutation of $\Bbb{N}$)" Please make your post clearer so that others find it easy to comment on or answer. $\endgroup$ Jun 6 at 11:53
  • $\begingroup$ @ShubhrajitBhattacharya Hope its clear now $\endgroup$
    – algo
    Jun 6 at 12:21
  • $\begingroup$ Your solution is correct. $\endgroup$
    – freakish
    Jun 6 at 12:47

1 Answer 1

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Your solution is correct, but here are just some comments on your proof:

(1) The symbol $\infty$ stands for infinity. So for your second paragraph, you should write "there are infinitely many elements in $\mathbb{N}$", or, $|\mathbb{N}|=\infty$.

(2) On the fourth paragraph, you denote $x_2\in \mathbb{N}$. You should emphasize that $x_2\neq x_1$ so that you can construct a permutation $\sigma'$ later on in order to get a contradiction.

(3) The permutation $\sigma'$ should be written as $$\sigma'(x)= \begin{cases} x_2,& x=x_1;\\ x_1,& x=x_2;\\ x,& x\neq x_1,x_2;\\ \end{cases}$$

(4) Lastly, you may explain why you have the contradiction, which is because $\sigma'\in G$ implies that $\sigma'\in \langle S\rangle$ but $\sigma'(x_1)\neq x_1$, which is absurd due to your observation in third paragraph.

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