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

ABSTRACT

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 {AC_B(A)}, a strengthening of earlier known results.

KEYWORDS:

• Chermak-Delgado lattice
• Chermak-Delgado measure
• Chermak-Delgado subgroup
• dihedral group
• Frobenius group
• subgroup lattice
• ZM-group

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20D30
• Secondary: 20D60, 20D99

Acknowledgments

The authors are grateful to Professor I. M. Isaacs for providing the statement of Theorem 3 and an outline of the proof, and also to the reviewer for its remarks which improve the previous version of the paper.