= Diagonal action on a centerless group
{title2=$(a,b):t\mapsto a^{-1}tb$}
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 $N\trianglelefteq T$: 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.
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 $(a,b)\cdot t=atb^{-1}$; it has the same diagonal <stabilizer subgroup>.
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