# What are some techniques for embedding a finite group into $S_m$ for $m$ as small as possible?

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).

• 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 – Ben Grossmann Dec 31 '19 at 20:16
• In this thread there is a classification of those groups that absolutely need $n=|G|$. – Jyrki Lahtonen Dec 31 '19 at 20:21
• @JyrkiLahtonen Thanks! – Ovi Dec 31 '19 at 20:24
• @Omnomnomnom Thanks! – Ovi Dec 31 '19 at 20:29
• If $G$ has a normal subgroup $H$ with trivial centralizer in $G$, then $G$ embeds in $S_{|H|}$ by conjugation. – user615081 Dec 31 '19 at 21:29