On the Chermak–Delgado lattices of split metacyclic $p$-groups

Abstract

The Chermak–Delgado measure of a subgroup $H$ of a finite group $G$ is defined as $m_G\left(H\right)=\left|H\right|\left|C_G\left(H\right)\right|$. The subgroups with maximal Chermak–Delgado measure form a poset and corresponding lattice, known as the CD-lattice of $G$. We describe the symmetric nature of CD-lattices in general, and use information about centrally large subgroups to determine the CD-lattices of split metacyclic $p$-groups in particular. We also describe a rank-symmetric sublattice of the CD-lattice of split metacyclic $p$-groups.