19.111 (2018)

Do there exist infinite groups all of whose proper subgroups are cyclic of order $p$ that do not satisfy any nontrivial group identity except $x^p = 1$ and its consequences?

Note that there exist infinite groups all of whose proper subgroups are infinite cyclic that do not satisfy any nontrivial group identity (due to A. Olshanskii; see P. Zusmanovich, J. Algebra, 388 (2013), 268–286, Remark after Theorem 6.1).