What is the meaning of this subgroup of $S_{n}$?

Question

What does the group $H=\left\{\beta \in S_{n} | \beta(1)\in \left\{1,2\right\}, \beta(2)\in\left\{1,2\right\}\right\}$ look like?

Is this the set of permutations that sends position 1 to position 1 or 2, and position 2 to 1 or 2?

I need to show that this is a subgroup of $S_{n}$ but I'm not sure I understand the notation here. I want to use the one-step subgroup test to prove that $H\leq S_{n}$ and to do that I would consider some $\alpha,\beta \in H$ and proceed to show that $\alpha\beta^{-1}\in H$. But if I don't even know what $H$ is, it is hard to show this.

Answer 1

Of course you know what $H$ is - it is written right there.

• Clearly, the identity is $\in H$
• Let $\beta,\gamma\in H$. As $\beta(1)\in\{1,2\}$ and both $\gamma(1)$ and $\gamma(2)$ are $\in\{1,2\}$, it follows that $\gamma(\beta(1))\in\{1,2\}$. Similarly, $\gamma(\beta(2))\in\{1,2\}$. We conclude that $\gamma\circ\beta\in H$.

As every non-empty and closed under multiplication subset of a finite group is a subgroup, we are done.

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By the way, for $n\ge 2$, we can readily see that $H\cong S_2\times S_{n-2}$.