The action of $S_5$ on $6$ points

Question

I wanted to show that the symmetric group on $5$ points ($S_5$ as a subgroup of $S_6$) acting on $6$ points is not a transitive action.

Is this a valid group action?

It is clearly not transitive because there are more points to act on but I am not sure if this is considered as a valid example. thank you

Do observe that there exists also a transitive action of $S_5$ on six points. Namely, $S_5$ has six subgroups of order $5$, $H_1,H_2,\ldots,H_6$, and it acts on the set ${H_1,H_2,\ldots,H_6}$ by conjugation. For all $\sigma\in S_5$ and all $i$, we have that $\sigma H_i\sigma^{-1}$ is one of the $H_j$s. It is easy to see that this is also an action. It gives us a different way of looking at $S_5$ as a subgroup of $S_6$. This is somewhat unusual, and eventually leads to a construction of a non-inner automorphism of $S_6$. — Jyrki Lahtonen, Dec 01 '19 at 21:14

score 1 · Accepted Answer · answered Dec 01 '19 at 20:21

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Yes, it is a valid action, provided that $S_5$ fixes an element of the six points, WLOG the sixth one. Verifying that this is an action is routine (and follows from the definition of $S_5$).

answered Dec 01 '19 at 20:21

Shaun

44,997

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Exactly what I needed. Thank you – mandella Dec 01 '19 at 20:22
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You're welcome, @mandella :) – Shaun Dec 01 '19 at 20:23
Please don't forget to "accept" answers, @mandella, by clicking the checkmark $\checkmark$ next to them. (I've noticed you haven't accepted answers for a while.) – Shaun Dec 01 '19 at 20:29
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I had to wait 10 minutes to accept it :) – mandella Dec 01 '19 at 20:35

The action of $S_5$ on $6$ points

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