Characterizing translations on groups by cosets of their subgroups

On a group, constant functions and left translations by group elements map left cosets into left cosets for every subgroup. We determine classes of groups for which this property of preserving cosets characterizes constants and translations, e.g. finite non-abelian groups that are perfect, partitioned, primitive, or generated by elements of prime order. For certain classes of groups we construct other coset-preserving functions, in particular, power endomorphisms and functions defined in terms of the subgroup lattice. Here our motivation to investigate the connection between the properties of a group and the properties of its subgroup lattice comes from Universal Algebra; the tools are from Group Theory. This is joint work with Andras Pongracz (ELTE Budapest) and Gabor Horvath (University of Debrecen).