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Consider the symmetric group $ \langle S_{10}, ◦ \rangle$

Show that $H = \{ \sigma \in S_{10} : \sigma(5) = 5\}$ is a subgroup of $S_{10}$

By the subgroup test a subset H of G is a subgroup if and only if it satisfies the following:

  1. The identity $e$ of $G$ is in $H$
  2. If $h_1, h_2 \in H$ then $h_1h_2 \in H$
  3. If $h \in H$ then $h^{-1} \in H$

But I have no idea how to apply that to this example, not for a lack of trying. I have a test on group theory soon and really need help understanding this problem. Could someone please explain how to show one of the conditions and then at least I can have a fresh attempt at the others. Thanks.

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  • $\begingroup$ The identity of $G$ is the identity permutation, it does nothing to each element. In particular, it preserves 5, so the identity of G is in H. $\endgroup$
    – B. Mehta
    Nov 1, 2017 at 22:57

3 Answers 3

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HINTS..

$H$ describes permutations that leave the element $5$ unchanged (invariants).

Is the identity permutation one that belongs to $H$? It does leave $5$ unchanged..

Assume $\sigma$ and $\tau$ belong to $H$. What happens with $\sigma\tau$ with respect to $5$?

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We can view the elements of $S_{10}$ as permutations of the numbers $1,\ldots,10$. The condition $\sigma(5)=5$ means the permutation $\sigma$ does not move the number $5$.

  1. What is the identity permutation in $S_{10}$? In particular, what does it do to the number $5$? (Specifically, does $\sigma=e$ satisfy $\sigma(5)=5$?)
  2. If two permutations $h_1$ and $h_2$ do not move the number $5$, then what happens to $5$ if you apply permutation $h_2$ and then $h_1$?
  3. Each permutation $h$ has an inverse (basically, if $h$ moves the numbers in some way, then $h^{-1}$ moves them back to their original position). If $h$ does not move $5$, then does $h^{-1}$?
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If $\sigma \in H$ then what can you say about $\sigma^{-1} (5),$ given what you know about $\sigma $?
If $\sigma \in H$ and $\tau \in H$ what can you say about $\sigma \tau(5)= \sigma(\tau(5))$ given what you know about $\sigma $ and $\tau$?
What do you know about $e(5)$?

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