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For Which Finite Groups G is the Lattice ℒ(G) of Subgroups Gorenstein?

Published online by Cambridge University Press:  22 January 2016

Takayuki Hibi
Takayuki Hibi*
Affiliation:
Department of Mathematics Faculty of Science Nagoya University, Chikusa-ku, Nagoya 464, Japan
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Let G be a finite group and ℒ(G) the lattice consisting of all subgroups of G. It is well known that ℒ(G) is distributive if and only if G is cyclic (cf. [2, p. 173]). Moreover, the classical result of Iwasawa [8] says that ℒ(G) is pure if and only if G is supersolvable. Here, a finite lattice is called pure if all of maximal chains in it have same length and a finite group G is called supersolvable if ℒ(G) has a maximal chain which consists of normal subgroups of G.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 105 , March 1987 , pp. 147 - 151
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

[ 1 ] Baclawski, K., Cohen-Macaulay ordered sets, J. Algebra, 63, (1980), 226–258.Google Scholar
[ 2 ] Birkhoff, G., “Lattice Theory”, 3rd ed., Amer. Math. Soc. Colloq. Publ. No. 25, Amer. Math. Soc, Providence, R. L, 1967.Google Scholar
[ 3 ] Björner, A., Shellable and Cohen-Macaulay partially ordered sets, Trans. Amer. Math. Soc, 260 (1980), 159–183.Google Scholar
[ 4 ] Björner, A., A Garsia and R. Stanley, An introduction to Cohen-Macaulay partially ordered sets, Ordered sets, Rival, I. (ed.), D. Reidel Publishing Company, 1982, 583–615.Google Scholar
[ 5 ] Hall, M., “The Theory of Groups”, The Macmillian Company, 1959.Google Scholar
[ 6 ] Hall, P., A note on soluble groups, J. London Math. Soc, 3 (1928), 98–105.Google Scholar
[ 7 ] Hochster, M., Cohen-Macaulay rings, combinatorics and simplicial complexes, Proc. Second Oklahoma Ring Theory Conf. (March, 1976), Dekker, 1977, 171–223.Google Scholar
[ 8 ] Iwasawa, K., Über die endlichen Gruppen und die Verbände ihrer Untergruppen, J. Univ. Tokyo, 43 (1941), 171–199.Google Scholar
[ 9 ] Reisner, G., Cohen-Macaulay quotients of polynomial rings, Adv. in Math., 21 (1976), 30–49.Google Scholar
[10] Stanley, R., Supersolvable lattices, Algebra Universalis, 2 (1972), 197–217.Google Scholar
[11] Stanley, R., “Combinatorics and Commutative Algebra”, Progress in Math. Vol. 41, Birkhäuser, 1983.Google Scholar
[12] Suzuki, M., “Structure of a group and the structure of its lattice of subgroups”, Springer-Verlag, 1956.Google Scholar