Lattice-complemented is not transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., lattice-complemented subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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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 .
Example of the dihedral group
Let be the dihedral group of order eight; specifically:
Let be the center of : .
Let be the elementary Abelian subgroup generated by and , so .
- 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.