# Prove $\{\sigma\in S_\Bbb N\mid\sigma(x)= x\text{ for all but finitely many }x\in\Bbb{N}\}$ is not finitely generated

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?

• 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. Jun 6 at 11:53
• @ShubhrajitBhattacharya Hope its clear now
– algo
Jun 6 at 12:21
• Your solution is correct. Jun 6 at 12:47

(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.