### A Characterization of Groups Whose Lattices of Subgroups are n–M_{p+1} Chains for All Primes p

*Chawewan Ratanaprasert*

Department of Mathematics, Faculty of Science, Silpakorn University, Nakorn Pathom

#### Abstract

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 M_{m}by characterizing all integers m ≥ 3 such that there exists a group G whose Sub(G) is isomorphic to M_{m}and also have characterized all groups G whose Sub(G) is isomorphic to M_{m}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 M_{m}for all integers m ≥ 3 of modular lattices, the class of n–M_{p+1}chains for all primes p, and all n ≥ 1 and characterized all groups G whose Sub(G) is an n–M_{p+1}chain. It happens that G is a group whose Sub(G) is an n–M_{p+1}chain if and only if G is an abelian p-group of the form Z_{pn}× Z_{p}. Moreover, we can tell numbers of all subgroups of order pi for each 1 ≤ i ≤ n of the special class of p-groups.**Key Words**: Modular lattice; Lattice of subgroups; p-groupFull Text: PDF