A note on the Chermak–Delgado lattice of a finite group

Abstract

In this note we describe the structure of finite groups G whose Chermak-Delgado lattice is the interval

Groups whose Chermak-Delgado lattice is a subgroup lattice of an elementary abelian $p$-group

“…In the last years, there has been a growing interest in understanding this lattice (see e.g. [1][2][3][4][5][6], [8][9], [12][13][14][15][16][17][18]). Notice that a Chermak-Delgado lattice is always self-dual.…”

Finite groups with a certain number of values of the Chermak–Delgado measure

“…The starting point for our discussion is given by Corollary 3 of [16], which states that there is no finite non-trivial group G such that CD(G) = L(G). In other words, m G has at least two distinct values for every finite non-trivial group G. This leads to the following natural question:…”

Subnormality and the Chermak–Delgado lattice

“…(1) Showing that certain types of structures occur as Chermak-Delgado lattices for various group: chains [3], antichains [4], diamonds [2], all subgroups containing Z(G) in a group [12], or even the Chermak-Delgado lattice for certain families of groups [9], [11], etc. (2) Showing properties of the Chermak-Delgado lattice in general: the Chermak-Delgado lattice of a direct product is the direct product of the Chermak-Delgado lattices of the factors [5].…”

“…Taking the central product approach also allows us to prove a result of Tȃrnȃuceanu [12,Corollary 4]. Tȃrnȃuceanu's result occurs as a corollary to their classification of groups G satisfying CD(G) = [Z(G) ∶ G] L(G) .…”