Abelian groups whose subgroup lattice is the union of two intervals

Abstract

In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice G] is the union of these intervals.2000 Mathematics subject classification: primary 06C99, 20K10, 20K27.

Congruence preserving expansions of nilpotent algebras

“…Seeking a contradiction, we assume m 1 > m 2 . By [BC05], S then splits. Lemma 2.1 from [AM13] describes lattices that do split, but not strongly.…”

Breaking points in the poset of conjugacy classes of subgroups of a finite group

“…and the abelian groups G satisfying (3) have been determined in [1]. The above concepts can be naturally extended to other remarkable posets of subgroups of G, such us the posets C(G) and C(G) of cyclic subgroups and of conjugacy classes of cyclic subgroups of G, respectively.…”

“…2 for some x ∈ G this is a poset with the least element [1] = {1} and the greatest element [G] = {G}. We will prove that the cyclic p-groups of order at least p 2 and the generalized quaternion 2-groups are the unique finite groups G for which L(G) has breaking points.…”

Breaking points in centralizer lattices

“…Let G be a finite group and L(G) be the subgroup lattice of G. The starting point for our discussion is given by [2], where the proper nontrivial subgroups H of G satisfying the condition (1) for every X ∈ L(G) we have either X ≤ H or H ≤ X have been studied. Such a subgroup is called a breaking point for the lattice L(G), and a group G whose subgroup lattice possesses breaking points is called a BP-group.…”

“…and the abelian groups G satisfying (3) have been determined in [1]. The above concepts can be naturally extended to other remarkable posets of subgroups of G, and also to arbitrary posets.…”