%0 Journal Article
%T A Characterization of Groups Whose Lattices of Subgroups are n¨CMp+1 Chains for All Primes p
%A Chawewan Ratanaprasert
%J Silpakorn University Science and Technology Journal
%D 2009
%I 
%X 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 ¡Ý 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 ¡Ý 3 of modular lattices, the class of n¨CMp+1 chains for all primes p, and all n ¡Ý 1 and characterized all groups G whose Sub(G) is an n¨CMp+1 chain. It happens that G is a group whose Sub(G) is an n¨CMp+1 chain if and only if G is an abelian p-group of the form Zpn ¡Á Zp. Moreover, we can tell numbers of all subgroups of order pi for each 1 ¡Ü i ¡Ü n of the special class of p-groups.
%K Modular lattice
%K Lattice of subgroups
%K p-group
%U http://www.journal.su.ac.th/index.php/sustj/article/view/141/162