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It is well known, that any finite group of order $n$ is isomorphic to a subgroup of $S_n$. Let’s call a finite group $G$ incompressible iff it is not isomorphic to any subgroup of $S_{|G|-1}$ . Does there exist some sort of classification of incompressible groups?

What I currently know:

Any non-trivial incompressible group has non-trivial center

If the center of a group $G$ is trivial, then it acts faithfully by conjugation on $G \setminus \{e\}$.

If an incompressible group is non-trivially decomposed into a direct product of two its subgroups, it is isomorphic to $C_2 \times C_2$

One can construct a faithful action of $H \times K$ on $H \cup K$. It is defined as $(h, k)h_0 \mapsto hh_0$ and $(h, k)h_0 \mapsto kk_0$ for $h, h_0 \in H$, $k, k_0 \in K$.

$|H| + |K| \geq |H||K|$ iff either one of the groups is trivial, or both of them are isomorphic to $C_2$.

$C_2 \times C_2$ is the only possible group and indeed is not contained in $S_3$.

I also conjecture, that «direct product» in this statement can be replaced with «semidirect product», but do not know how to prove that.

All cyclic $p$-groups are incompressible

If $p$ is prime, then $S_{p^n - 1}$ does not have an element of order $p^n$

$Q_8$ is incompressible

$S_7$ does not contain $Q_8$ as a subgroup

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    $\begingroup$ According to Jack Schmidt's answer to mathoverflow.net/questions/16858, the only examples are $C_2\times C_2$, cyclic groups of prime power order, and (generalized) quaternion groups. He gives a reference to a paper by D.L Johnson. $\endgroup$ – Derek Holt Jun 4 at 8:51
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    $\begingroup$ You could have considered another word for the definition. 😂 $\endgroup$ – MathematicsStudent1122 Jun 4 at 9:06
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    $\begingroup$ Why "horny"?!?! $\endgroup$ – user1729 Jun 4 at 9:16
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    $\begingroup$ At the risk of being boring, I have seen such groups be called 'incompressible' $\endgroup$ – Robert Chamberlain Jun 4 at 9:47
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    $\begingroup$ Well I dislike "horny" because I find it distracting. $\endgroup$ – Derek Holt Jun 4 at 11:31
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These were fully classified by Johnson in the paper 'Minimal Permutation Representations of Finite Groups'.

A group is incompressible iff it is isomorphic to one of the following:

  • Cyclic group of prime power order $C_{p^n}$
  • Generalised quaternion $2$-group $\langle x,y|x^{2^n}=1,x^{2^{n-1}}=y^2,x^y=x^{-1}\rangle$
  • the Klein four-group $C_2\times C_2$

The proof is reasonably short so well worth looking up!

Reference: Johnson, D. L. "Minimal permutation representations of finite groups." Amer. J. Math. 93 (1971), 857-866. MR 316540 DOI: 10.2307/2373739.

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  • $\begingroup$ What does $x^y$ mean in this context? $\endgroup$ – celtschk Oct 17 at 22:01
  • $\begingroup$ Here $x^y=y^{-1}xy$ $\endgroup$ – Robert Chamberlain Oct 17 at 22:08
  • $\begingroup$ So the last condition is equivalent to $xyx=y$, right? $\endgroup$ – celtschk Oct 17 at 22:14
  • $\begingroup$ the last condition for the generalised quaternion $2$-group, yes $\endgroup$ – Robert Chamberlain Oct 18 at 7:54

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