Abstract. This paper provides the first steps in classifying the finite solvable groups having Property A, which is a property involving abelian normal subgroups. We see that this classification is reduced to classifying the solvable Chermak-Delgado simple groups, which the author defines. The author completes a classification of Chermak-Delgado simple groups under certain restrictions on the primes involved in the group order.

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.

Abstract. The Chermak-Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of G. The least element of the Chermak-Delgado lattice of G is known as the Chermak-Delgado subgroup of G. This paper concerns groups with a trivial Chermak-Delgado subgroup. We prove that if the Chermak-Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak-Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak-Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak-Delgado subgroup. We establish lattice theoretic properties of Chermak-Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain p-group lattices play a major role in the author's constructions.

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result to classify the Chermak-Delgado lattices of dicyclic groups and of metabelian p-groups of maximal class.

No abstract