Non-commutative simple group operates non-trivially on a set with less than $5$ elements

Question

Either prove or disprove that a non-commutative simple group can or cannot operate non-trivially on a set with less than $5$ elements.

What does the term "operate on a set" mean? And also "operate trivially"?

Answer 1

Suppose that $\rho : G\longrightarrow {\rm Aut}(X)$ is a group action with $G$ simple. If $\rho$ is nontrivial, the fact that $G$ is simple means $\ker\rho=0$, that is, the action is faithful. If $|X|=n$; then $G$ is a subgroup of ${\rm Aut}(X)$, which has $n!$ elements. Now if $n<5$; then $n!=1,2,6,24$, so that the order of $G$ is a divisor of $24$. But the smallest nonabelian simple group has order $5!/2=60$.

Answer 2

It probably means a group action.

The symmetric group $S_3$ (containing all permutations of $3$ elements) is noncommutative and acts on $3$ elements nontrivially, which means that not every group element performs the identity action.
However, it is not simple, so we're not there yet.

A further hint: see the list of finite simple groups