By simple permutations and by fixing one element say $f(6)=6$,we can get a copy of $S_5$ in $S_6$ . Similarly we can obtain 6 different copies , can I obtain a 7th copy?
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By Copy I mean $S_5$ is isomorphic to a subgroup of $S_6$ obtained by fixing one element and permutating the others – Raunak Banerjee Feb 02 '20 at 07:03
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Then it is a simple matter of counting the number of ways we can choose k out of n elements to fix. So its a simple combination to see the number of subgroups of size k are isomorphic to $S_k$ – Edcookie274 Feb 02 '20 at 07:06
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3@DonThousand $S_6$ has outer automorphisms which, if applied to one of the "normal" copies of $S_5$ (fixing a point), yields an "exotic" copy of $S_5$ (does not fix a point). – Ted Feb 02 '20 at 08:09
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@Ted can you elaborate, I didn't get it – Antimony Feb 03 '20 at 14:54