J-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

Definition

A group is termed a J-group if its lattice of subgroups satisfies the Jordan-Dedekind chain condition: any two maximal chains between two subgroups have the same length.

For finite groups, this is equivalent to the property of being supersolvable, and hence, a finite supersolvable group.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes		Suppose $G$ is a J-group and $H$ is a subgroup of $G$. Then, $H$ is also a J-group.
quotient-closed group property	Yes		Suppose $G$ is a J-group and $H$ is a normal subgroup of $G$. Then, the quotient group $G/H$ is also a J-group.
lattice-determined group property	Yes		Whether or not a group is a J-group can be determined completely from its lattice of subgroups. In particular, if $G_{1},G_{2}$ have isomorphic lattices of subgroups, then $G_{1}$ is a J-group if and only if $G_{2}$ is a J-group.