Groups whose Chermak–Delgado lattice is a chain

Abstract

Abstract. For a finite group G with subgroup H, the Chermak–Delgado measure of H in G refers to | H | | C G ( H ) | ${\vert H \vert \,\vert C_G(H) \vert }$ . The set of all subgroups with maximal Chermak–Delgado measure form a sublattice, 𝒞𝒟 ( G ) $\mathcal {CD}(G)$ , within the subgroup lattice of G. This paper examines conditions under which the Chermak–Delgado lattice is a chain of subgroups H 0 < H 1 < ⋯ < H n ${H_0 < H_1 < \cdots < H_n}$ . On the basis of a general result about extending certain Chermak–Delgado lattices, we construct, for any prime p and any non-negative integer n, a p-group whose Chermak–Delgado lattice is a chain of length n.