Diagonal action on a centerless group ($(a,b):t\mapsto a^{-1}tb$)

For nontrivial centerless $T$, the displayed right group 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 $N⊴T$: 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.

The convention is right-handed: writing $t^{(a,b)}=a^{-1}tb$, one has $(t^{(a,b)}{)}^{(c,d)}=t^{(ac,bd)}$. With left group action notation, the corresponding formula is $(a,b)\cdot t=at{b}^{-1}$; it has the same diagonal stabilizer subgroup.