Metacyclic p-groups and their conjugacy classes of subgroups

摘要

Let G be a group and let ℓ(G) be the set of all conjugacy classes H of subgroups H of G, where a partial order ≤ is defined by H1 ≤ H2 if and only if H1, is contained in some conjugate of H2.A number of papers (see for example 1 and the references mentioned there) deal with the question of characterizing groups G by the poset ℓ(G). For example, in 1 it was shown that if ℓ(G) and ℓ(H) are order-isomorphic and G is a noncyclic p-group then G = H. Moreover, if G is abelian, then G = H, and if G is metacyclic then H is metacyclic.