# For which $n$ is the symmetric group $S_n$ a subgroup of the special orthogonal group $SO(3)$?

For which $$n$$ is the symmetric group $$S_n$$ a subgroup of the special orthogonal group $$SO(3)$$? For example, this holds for $$n≤4$$, however I don't know if it holds for $$n=5$$ or what happens for larger $$n$$.

• It would be a faithful $3$-dimensional representation which doesn't exist for $n\ge 5$ Dec 26, 2020 at 1:29

This question can be attacked either as a question about the finite subgroups of $$SO(3)$$ or as a question about the representation theory of $$S_n$$. Each of these approaches will prove that $$S_n \hookrightarrow SO(3)$$ iff $$n \le 4$$.

Finite subgroups: It's a classical result that the complete list of finite subgroups of $$SO(3)$$ is the following:

• the cyclic groups $$C_n$$,
• the dihedral groups $$D_n$$,
• the tetrahedral group $$A_4$$,
• the octahedral group $$S_4$$, or
• the icosahedral group $$A_5$$.

$$S_4$$ appears on this list but $$S_n$$ does not for $$n \ge 5$$.

Representation theory: The symmetric group $$S_n$$ is known to have the property that the lowest-dimensional faithful representation has dimension $$n-1$$ for $$n \ge 5$$, as reuns points out in the comments.