On Diamond-Free Subgroup Lattices

TL;DR

This paper introduces the concept of cyclic-diamond sublattices within subgroup lattices and characterizes their presence in finite and infinite groups, revealing structural properties of these groups.

Contribution

It defines cyclic-diamond lattices and proves their unavoidable presence in finite non-cyclic groups, while characterizing their absence in certain infinite abelian groups.

Findings

Finite non-cyclic groups always contain a cyclic-diamond sublattice.

Infinite abelian groups lack cyclic-diamond sublattices if all finitely generated subgroups are cyclic or isomorphic to Z×Z_{2^N}.

Provides a structural criterion for the presence of cyclic-diamond sublattices.

Abstract

In this paper we introduce a particular lattice of subgroups called a "cyclic-diamond" and show that every finite non-cyclic group contains a cyclic-diamond as a sublattice of its lattice of subgroups. Turning to the infinite case, we show that an infinite abelian group does not contain a cyclic-diamond in its subgroup lattice if and only if all of its finitely generated subgroups are cyclic or isomorphic to $\mathbb{Z} \times \mathbb{Z}_{2^N}$ for some $N$.