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A Characterization of Groups Whose Lattices of Subgroups are nMp+1 Chains for All Primes p |
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| รหัสดีโอไอ | |
| Creator | Chawewan Ratanaprasert |
| Title | A Characterization of Groups Whose Lattices of Subgroups are nMp+1 Chains for All Primes p |
| Publisher | Silpakorn University Research and Development Institute |
| Publication Year | 2552 |
| Journal Title | Silpakorn University Science and Technology Journal |
| Journal Vol. | 3 |
| Journal No. | 2 |
| Page no. | 42-47 |
| Keyword | Modular lattice, Lattice of subgroups, p-group |
| ISSN | 1905-9159 |
| 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 Mm by characterizing all integers m ? 3 such that there exists a group G whose Sub(G) is isomorphic to Mm and also have characterized all groups G whose Sub(G) isisomorphic to Mm forsome 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 nMp+1 chains for all primes p, and all n ? 1 and characterized all groups G whose Sub(G) is an nMp+1 chain. It happens that G is a group whose Sub(G) is an nMp+1 chain if and only if G is an abelian p-group of the form n Z Z p p ? . Moreover, we can tell numbers of all subgroups of order pi for each 1 ? i ? n of the special class of p-groups. |
