Every synchronising permutation group is primitive and of one three types: affine, almost simple, or diagonal. We exhibit the first known example a diagonal type group. More precisely, we show that PSL ( 2 , q ) × $\operatorname{{\mathrm{PSL}}}(2,q)\times \operatorname{{\mathrm{PSL}}}(2,q)$ acting in its action on $\operatorname{{\mathrm{PSL}}}(2,q)$ separating, hence synchronising, for = 13 $q...

Our main goal is to study the freeness of Schützenberger groups defined by primitive substitutions. findings include a simple test for these groups, which applied exhibit invertible substitution with corresponding non-free group. This constitutes counterexample result Almeida dating back 2005. We also give some early results concerning relative question remains largely unexplored.