#### Vol. 9, No. 5, 2016

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

### Erin Brush, Jill Dietz, Kendra Johnson-Tesch and Brianne Power

Vol. 9 (2016), No. 5, 765–782
##### Abstract

The Chermak–Delgado measure of a subgroup $H$ of a finite group $G$ is defined as ${m}_{G}\left(H\right)=|H||{C}_{G}\left(H\right)|$. 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.

##### Keywords
centrally large subgroups, Chermak–Delgado measure, lattices of subgroups, metacyclic $p$-groups
Primary: 20D30