# J-group

From Groupprops

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 propertiesVIEW 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 is a J-group and is a subgroup of . Then, is also a J-group. | |

quotient-closed group property | Yes | Suppose is a J-group and is a normal subgroup of . Then, the quotient group 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 have isomorphic lattices of subgroups, then is a J-group if and only if is a J-group. |