Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero

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It is an open question in the study of Chermak-Delgado lattices precisely which finite groups G have the property that CD(G) is a chain of length 0. In this note, we determine two classes of groups with this property. We prove that if G=AB is a finite group, where A and B are abelian subgroups of relatively prime orders with A normal in G, then the Chermak-Delgado lattice of G equals {ACB(A)}, a strengthening of earlier known results.