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 properties
VIEW 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: (facts closely related to K-group, all facts related to K-group) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
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 ![]() ![]() ![]() ![]() ![]() ![]() |
quotient-closed group property | Yes | Suppose ![]() ![]() ![]() ![]() | |
finite direct product-closed group property | Yes | Suppose ![]() ![]() | |
lattice-determined group property | Yes | Given two groups ![]() |
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 |