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I know that if $|G| = n$, then $G$ can be embedded into $S_n$. But the group $S_n$ is very large compared to $G$, so I was wondering if there are general ways of embedding $G$ into a smaller symmetric group. (By general I don't mean that it has to work for all groups, but hopefully for large classes of groups)

Also, I was wondering embedding groups into smaller symmetric groups is common/useful, or just a curiosity?

The only approach I could think of is to let $G$ act on various things, and hope that the action is faithful. One nice thing, for example, is that if $G$ has a simple subgroup $H$ (which is itself non-normal in $G$), then the action of $G$ on the left cosets of $H$ is faithful, since according to this question, the kernel must be trivial.

Furthermore, if you let $G$ act by conjugation on a subgroup, I believe the action is sometimes faithful, sometimes not (has to do with weather the conjugates are disjoint).

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    $\begingroup$ Note for instance that representing a cyclic group of order $n$ would require an $S_m$ such that $g(m) \geq n$, where $g$ denotes Landau's function $\endgroup$ – Ben Grossmann Dec 31 '19 at 20:16
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    $\begingroup$ In this thread there is a classification of those groups that absolutely need $n=|G|$. $\endgroup$ – Jyrki Lahtonen Dec 31 '19 at 20:21
  • $\begingroup$ @JyrkiLahtonen Thanks! $\endgroup$ – Ovi Dec 31 '19 at 20:24
  • $\begingroup$ @Omnomnomnom Thanks! $\endgroup$ – Ovi Dec 31 '19 at 20:29
  • $\begingroup$ If $G$ has a normal subgroup $H$ with trivial centralizer in $G$, then $G$ embeds in $S_{|H|}$ by conjugation. $\endgroup$ – user615081 Dec 31 '19 at 21:29
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This is a very broad question and an active area of study, but two papers I found useful to start with were Minimal Permutation Representations Of Finite Groups by Johnson and The minimal degree of permutation representations of finite groups by Becker.

The first of these is one of the earlier papers I've read, so doesn't assume much knowledge and the second is, I think, a masters thesis so is excellent for a less experienced reader.

There are dozens of other papers you could read and many MSE posts, the key phrase to search is 'minimal permutation representations' and you are likely to stumble across representation theory as permutation representations are often studied as matrix representations.

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    $\begingroup$ Although too advanced for my level, perhaps you can appreciate also "On minimal faithful permutation representations of finite groups", David Easdown, Cheryl E Praeger, Bulletin of the Australian Mathematical Society 38 (2), 207-220, 1988. $\endgroup$ – user615081 Jan 1 '20 at 10:05

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