Lattice-complemented subgroup

Symbol-free definition
A subgroup of a group is said to be lattice-complemented if there is another subgroup such that:


 * The two subgroups intersect trivially
 * The join of the two subgroups is the whole group

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be lattice-complemented if there is another subgroup $$K$$ such that:


 * $$H\cap K$$ is trivial
 * $$\langle H,K \rangle = G$$

Stronger properties

 * Weaker than::Direct factor
 * Weaker than::Complemented normal subgroup
 * Weaker than::Retract
 * Weaker than::Permutably complemented subgroup
 * Weaker than::Free factor

Metaproperties
A lattice-complemented subgroup of a lattice-complemented subgroup need not be lattice-complemented.

If $$H$$ is a lattice-complemented subgroup of a group $$G$$, and $$H \le L \le G$$, $$H$$ is not necessarily lattice-complemented in $$L$$.