  We show that the modular isomorphism problem has a positive answer for groups
of nilpotency class 2 with cyclic center, i.e. that for such p-groups G and H
an isomorphism between the group algebras FG and FH implies an isomorphism of
the groups G and H for F the field of p elements. For groups of odd order this
implication is also proven for F being any field of characteristic p. For
groups of even order we need either to make an additional assumption on the
groups or on the field.
