Lattice-complemented is not transitive
From Groupprops
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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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.