Y. Berkovich formulated the following problem: «Suppose that p-group (! satisfies the following
condition: if C is a cyclic subgroup o f G then either ('< Z(G) or C n Z (C ) = {l}. Classify all such
groups.» The authors have proved that each regular p-group satisfies this condition if and only if it is
either abelian or a group o f exponent p.
Here we regard the non-regular 2-groups. Some necessary conditions for non-abe/ian 2-group in which
every cyclic subgroup is contained in the center or has a trivial intersection with it are pointed. The
minima! example o f non-regular 2-group with 2 generators satisfying this condition is constructed.