Abstract

The paper classifies those locally finite groups having a proper nontrivial subgroup which is comparable with any other element of the subgroup lattice.

Introduction

Let G be a group and let L(G) denote its subgroup lattice. The description of groups G with L(G) a chain is well-known. In a chain, every element is comparable with the others. This raises the natural question of seeing what can be said about groups G having a proper nontrivial subgroup H with the property that for every subgroup X of G one has either X ≤ H or H ≤ X. Such a subgroup H will be called a breaking point for the lattice L(G). For the sake of convenience, we shall call these groups BP-groups.

Of course, BP-groups cannot be decomposed as nontrivial direct products. Moreover, if G is a BP-group with breaking point H, then every subgroup K of G strictly containing H is itself a BP-group with breaking point H. These simple considerations are valuable in what follows and we shall use them without any further reference.

Standard results from abelian group theory dispose of the structure of abelian BP-groups: these are cyclic p-groups in the finite case and Prüfer p-groups Z(p∞) in the infinite case. This focuses the discussion on nonabelian BP-groups.

As more exotic examples, the so-called extended Tarski groups, see Ol'shanskii [3], p. 344 are also BP-groups.