A Characterization of Groups Whose Lattices of Subgroups are n–Mp+1 Chains for All Primes p

Whitman, P.M. and Birkhoff, G. answered a well-known open question that for each lattice L there exists a group G such that L can be embedded into the lattice Sub(G) of all subgroups of G. Gratzer, G. has characterized that G is a finite cyclic group if and only if Sub(G) is a finite distributive lattice. Ratanaprasert, C. and Chantasartrassmee, A. extended a similar result to a subclass of modular lattices Mm by characterizing all integers m ≥ 3 such that there exists a group G whose Sub(G) is isomorphic to Mmand also have characterized all groups G whose Sub(G) is isomorphic to Mm for some integers m. On the other hand, a very well-known open question in Group Theory asked for the number of all subgroups of a group. In this paper, we consider the extension of the subclass Mm for all integers m ≥ 3 of modular lattices, the class of n–Mp+1 chains for all primes p, and all n ≥ 1 and characterized all groups G whose Sub(G) is an n–Mp+1 chain. It happens that G is a group whose Sub(G) is an n–Mp+1 chain if and only if G is an abelian p-group of the form Zpn × Zp. Moreover, we can tell numbers of all subgroups of order pi for each 1 ≤ i ≤ n of the special class of p-groups.