The base size of Aut(R) on the subgroup lattice of R equals 1 if and only if R is cyclic.
Babai,On the abstract group of automorphisms, Combinatorics, Proceedings 8th British Combinatorics Conference, Swansea 1981, ed
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On the base size of a finite group on its action on the lattice of subgroups
The base size of Aut(R) on the subgroup lattice of R equals 1 if and only if R is cyclic.