Metacyclic p-groups and their conjugacy classes of subgroups

Abstract

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.

The Posets of Classes of Isomorphic Subgroups of Finite Groups

“…It is an usual technique to consider an equivalence relation ∼ on an algebraic structure and then to study the factor set with respect to ∼, partially ordered by certain ordering relations. In the case of subgroup lattices, one of the most significant example is the poset C(G) of conjugacy classes of subgroups of a group G (see [2,3,4] and [9,10]). The current paper deals with the more general equivalence relation on the subgroup lattice of G induced by isomorphism.…”

On the Isomorphism Problem for Burnside Rings

“…First of all, the groups G and H have been shown in [3] to have isomorphic posets of conjugacy classes, so that they were natural candidates to try. The tables of marks for G and H were calculated using the following sequence of GAP commands: Clearly, a permutation n of the conjugacy classes of subgroups of H does not affect the ring structure of £l(H), and our problem was reduced to finding a suitable permutation matrix P related to n such that PTM(H)P = M(G).…”

“…The latter contains much less information about G. Indeed, all groups of order pq have order-isomorphic posets of conjugacy classes. However, it was shown in [1] that ^(G) = ^(H) and H a noncyclic p-group implies that |G| = \H\ and if H is abelian or metacyclic, then G and H are isomorphic (see [1] and [3]). Also, the result proved in [2] can be viewed in this context.…”

Untitled

“…Recently many authors have investigated the number k(G) of conjugacy classes of a group G. There are several papers on the conjugacy classes of finite p-groups [1][2][3] . Many authors obtained significant results but only on the lower and upper bound of k(G).…”