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

Abstract

Whitman, P.M. and Birkhoff, G. answered a well-known open question that for each lattice L thereexists a group G such that L can be embedded into the lattice Sub(G) of all subgroups of G. Gratzer, G. hascharacterized 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 M_m by characterizing allintegers m 3 such that there exists a group G whose Sub(G) is isomorphic to Mm and also have characterizedall groups G whose Sub(G) is isomorphic to M_m for some integers m. On the other hand, a very well-known openquestion in Group Theory asked for the number of all subgroups of a group. In this paper, we consider theextension of the subclass M_m for all integers m ³ 3 of modular lattices, the class of n–M_p+1 chains for all primesp, and all n ³ 1 and characterized all groups G whose Sub(G) is an n–M_p+1 chain. It happens that G is a groupwhose Sub(G) is an n–M_p+1 chain if and only if G is an abelian p-group of the form pn p Z _´Z _. Moreover, wecan tell numbers of all subgroups of order p_i for each 1 i n of the special class of p-groups.