Progression-free sets in class $2$ and arbitrary finite groups

We study progression-free sets in arbitrary finite groups. We transfer sharp upper bounds from abelian groups to nilpotent groups of class $2$, and from nilpotent groups to arbitrary groups. Combining a recent result of Raghavan with the author's theorem on large subgroups of class $2$ gives \[ r_3(G) \, ≤ \, C \, |G|exp( -c{(log|G|)1/6}{(loglog|G|)1/3}) \] for every finite group as |G|→∞. This improves the estimates obtained by using the abelian subgroup supplied by Pyber's theorem.