# K-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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## Contents

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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View a list of other standard non-basic definitions

## 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