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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

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