K-group

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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.
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Definition

A group $G$ is termed a K-group if every subgroup of the group is lattice-complemented. Explicitly, this means that given any subgroup $H$ of $G$, there is a subgroup $L$ of $G$ such that the intersection of subgroups $H \cap L$ is trivial and the join of subgroups $H$ and $L$ (namely, $\langle H,L \rangle$) is $G$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	See next column	It is possible to have a group $G$ and a subgroup $H$ such that $G$ is a K-group but $H$ is not. For instance, take $G$ to be symmetric group:S4 and $H$ to be D8 in S4 or Z4 in S4.
quotient-closed group property	Yes		Suppose $G$ is a K-group, and $H$ is a normal subgroup of $G$. Then, the quotient group $G/H$ is also a K-group.
finite direct product-closed group property	Yes		Suppose $G_1, G_2, \dots, G_n$ are all K-groups. Then, the external direct product $G_1 \times G_2 \times \dots \times G_n$ is also a K-group.
lattice-determined group property	Yes		Given two groups $G_1, G_2$ 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

See also

• Solvable K-group