Vol. 9, No. 5, 2016

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ISSN: 1944-4184 (e-only)
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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 mG(H) = |H||CG(H)|. 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
Mathematical Subject Classification 2010
Primary: 20D30
Milestones
Received: 16 March 2015
Revised: 1 October 2015
Accepted: 27 October 2015
Published: 25 August 2016

Communicated by Kenneth S. Berenhaut
Authors
Erin Brush
St. Olaf College
Northfield, MN 55057
United States
Jill Dietz
Department of Mathematics, Statistics and Computer Science
St Olaf College
1520 St. Olaf Avenue
Northfield, MN 55057
United States
Kendra Johnson-Tesch
St. Olaf College
Northfield, MN 55057
United States
Brianne Power
St. Olaf College
Northfield, MN 55057
United States