Breaking points in subgroup lattices

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

“…Clearly, all cyclic p-groups of order at least p 2 and all generalized quaternion 2-groups Q 2 n = a, b | a 2 n−2 = b 2 , a 2 n−1 = 1, b −1 ab = a −1 , n ≥ 3, are BP-groups. Note that a complete classification of BP-groups can be found in [2]. Also, we observe that the condition (1) is equivalent to…”

“…By bringing together Theorem 1 and the main results of [2,3,7], one obtains: In what follows we will denote by C the class of finite groups G satisfying Clearly, (4) is equivalent with (3) for abelian groups G because in this case we have L(G) = L(G). Then all finite abelian groups satisfying (3) belong to C. So, we may restrict our attention only to finite non-abelian groups contained in C. By Theorem 1, we know that the generalized quaternion 2groups Q 2 n , n ≥ 3, have this property.…”

Abelian groups whose subgroup lattice is the union of two intervals

“…Although the following result was already stated in [1], we supply a specific 'abelian' proof. Hence G/pH is cocyclic and, having elements of order p (in H/pH), must be a pgroup.…”

“…In [1], an arbitrary group G was called a BP-group if it has a proper subgroup H such that the subgroup lattice L{G) is the union of the intervals [1, H] and [//, G] (that is, any subgroup of G is either contained in H or contains H). The subgroup H was called a breaking point for the lattice L{G).…”

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.…”