# K-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

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to K-group, all facts related to K-group) |Survey articles about this | Survey articles about definitions built on this

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View a list of other standard non-basic definitions

## Definition

A group is termed a **K-group** if every subgroup of the group is lattice-complemented. Explicitly, this means that given any subgroup of , there is a subgroup of such that the intersection of subgroups is trivial and the join of subgroups and (namely, ) is .

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | No | See next column | It is possible to have a group and a subgroup such that is a K-group but is not. For instance, take to be symmetric group:S4 and to be D8 in S4 or Z4 in S4. |

quotient-closed group property | Yes | Suppose is a K-group, and is a normal subgroup of . Then, the quotient group is also a K-group. | |

finite direct product-closed group property | Yes | Suppose are all K-groups. Then, the external direct product is also a K-group. | |

lattice-determined group property | Yes | Given two groups that have isomorphic lattices of subgroups, either both are K-groups, or neither is. |

## Facts

If a core-free maximal subgroup of a group is a K-group, so is the whole group. Note that groups that have core-free maximal subgroups are called primitive groups.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notinos |
---|---|---|---|---|

finite simple group | finite and simple: no proper nontrivial normal subgroup | finite simple implies K (nontrivial proof, relies on classification of finite simple groups) | |FULL LIST, MORE INFO | |

C-group | every subgroup is a permutably complemented subgroup | alternating group:A4 is a K-group but not a C-group | |FULL LIST, MORE INFO | |

SK-group | every subgroup is a K-group | |FULL LIST, MORE INFO | ||

SC-group | every subgroup is a C-group | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Frattini-free group | Frattini subgroup (intersection of maximal subgroups) is trivial | K implies Frattini-free | |FULL LIST, MORE INFO |