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Recall the following classic result:

Cayley's Theorem Let $G$ be a group, and let $\operatorname{Perm}(G)$ be the group of set-theoretic bijections $G \to G$. The function $g \mapsto (h \mapsto gh) : G \to \operatorname{Perm}(G)$ is an injective group homomorphism.

Corollary Let $G$ be a finite group of order $n$. Then $G$ is isomorphic to a subgroup of $S_n$.

One may ask if Cayley's Theorem is "sharp", i.e. is it possible to embed a finite group $G$ in a smaller symmetric group than $S_{\lvert G \rvert}$? Let's say that a group $G$ is sharp if $G$ does not embed in $S_n$ for any $n < \lvert G \rvert$.

My question is: which finite groups are sharp?

This seems hard to answer in general but I'd be interested to learn what (partial) results are known. Here are some quick observations:

  • Let $G$ be a cyclic group of order $p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$ with $p_1, \dots, p_k$ being distinct primes and each $a_i \geq 1$. Then $G$ embeds in $S_n$ with $n = p_1^{a_1} + p_2^{a_2} + \cdots + p_k^{a_k}$. If $k > 1$ then $n < \lvert G \rvert$, so $G$ is not sharp.

  • The quaternion group (of order $8$) is sharp.

  • If $n > 2$ then $S_n$ is not sharp, since it embeds in itself and $n < n!$

It also makes sense to ask if infinite groups are sharp (with the same definition), and I'd be interested to hear results there as well! Obviously, any countably infinite group is sharp.

diracdeltafunk
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    Your first bullet point can be generalized to $G=A\times B$ with $|A|\ge 2$, $|B|\ge 3$. – Hagen von Eitzen Oct 17 '19 at 20:03
  • Trivially, cyclic groups of prime orders are sharp. Perhaps you could add this into your bullet points. As Hagen von Eitzen suggests, all finite decomposable groups are not sharp except $C_2^2$, where $C_2$ is the cyclic group of order $2$. – Batominovski Oct 17 '19 at 20:35

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