Diagonal action on a centerless group
= Diagonal action on a centerless group
{title2=$(a,b):t\mapsto a^{-1}tb$}
For nontrivial centerless $T$, this action of $T\times T$ is faithful and transitive. Its identity stabilizer is the diagonal subgroup. Subgroups above that diagonal are $\{(a,b):a^{-1}b\in N\}$ with $N\trianglelefteq T$: multiply by a diagonal element to isolate the second coordinate, then use diagonal conjugation. Thus the action is primitive exactly when $T$ is nonabelian simple.