J-group

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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.