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We show that on the four-symbol full shift, there is a finitely generatedsubgroup of the automorphism group whose action is (set-theoretically)transitive of all orders on the points of finite support, up to the necessarycaveats due to shift-commutation. As a corollary, we obtain that there is afinite set of automorphisms whose centralizer is $\mathbb{Z}$ (the shiftgroup), giving a finitary version of Ryan's theorem (on the four-symbol fullshift), suggesting an automorphism group invariant for mixing SFTs. We showthat any such set of automorphisms must generate an infinite group, and alsoshow that there is also a group with this transitivity property which is asubgroup of the commutator subgroup and whose elements can be written ascompositions of involutions. We ask many related questions and prove some easytransitivity results for the group of reversible Turing machines, topologicalfull groups and Thompson's~$V$.

Transitive action on finite points of a full shift and a finitary Ryan's theorem