K-group

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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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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