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. MSC2000 : Primary 20D30; Secondary 20D60, 20D99.

On the Chermak–Delgado lattice of a finite group

“…In this section we present a result which generalizes Proposition 7 in [10]. It will be used both in Sections 3 and 4.…”

“…The following appears in [10]. Finally, we present a proposition also applicable to describing CD(Dic 4n ).…”

“…In the last years there has been a growing interest in understanding this lattice (see e.g. [1,2,3,4,6,8,9,10,13,14,15,18]). Recall two important properties of the Chermak-Delgado lattice that will be used in our paper:…”

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

“…In the last years there has been a growing interest in understanding this lattice (see e.g. [3,4,5,6,8,10,11,12,13,15,18,20]). We recall several important properties of the Chermak-Delgado measure that will be used in our paper:…”