# Lattice-complemented is not transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., lattice-complemented subgroup)notsatisfying a subgroup metaproperty (i.e., transitive subgroup property).

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## Statement

It can happen that we have subgroups such that is a lattice-complemented subgroup of and is a lattice-complemented subgroup of but is *not* lattice-complemented in .

## Related facts

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8, Klein four-subgroups of dihedral group:D8, center of dihedral group:D8, subgroup structure of dihedral group:D8`

Let be the dihedral group of order eight; specifically:

.

Let be the center of : .

Let be the elementary Abelian subgroup generated by and , so .

We have:

- is lattice-complemented in : The subgroup is a permutable complement to in , and in particular, a lattice complement to in .
- is lattice-complemented in : The subgroup is a permutable complement to in , and in particular, is a lattice complement to in .
- is not lattice-complemented in : This can be seen by inspection, but it also follows from a more general fact about nilpotent groups: every nontrivial normal subgroup of a nilpotent group intersects the center nontrivially. A lattice-complement to the center must be a permutable complement, hence must be a nontrivial normal subgroup, and hence such a thing cannot exist in a nilpotent group.