Groups with elementary abelian commutant of at most $p^2$th order

Abstract

We obtain a representation of nilpotent groups with a commutant of the type $(p)$ or $(p, p)$ that has the form of a product of two normal subgroups. One of these subgroups is constructively described as a Chernikov $p$-group of rank 1 or 2, and the other subgroup has a certain standard form. We also obtain a representation of nonnilpotent groups with a commutant of the type $(p)$ or $(p, p)$ in the form of a semidirect product of a normal subgroup of the type $(p)$ or $(p, p)$ and a nilpotent subgroup with a commutant of order $p$ or 1.