No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined
It is possible to have a group , a lattice automorphism of the lattice of subgroups, and subgroups of with , such that:
- is a normal Sylow subgroup but not a Sylow direct factor (and hence, not a retract).
- is a Sylow retract but not a normal Sylow subgroup.
No subgroup property of any of the following kinds is a conditionally lattice-determined subgroup property:
- Any property equal to or weaker than normal Sylow subgroup but stronger than or equal to subnormal subgroup. This includes the properties: fully invariant subgroup, characteristic subgroup, normal subgroup, normal Hall subgroup, complemented normal subgroup.
- Any property equal to or weaker than Sylow retract but stronger than or equal to retract. This includes the properties Sylow retract, Hall retract, and retract.
- There exists an abelian group of prime power order that is lattice-isomorphic to a non-abelian group not of prime power order
Choose primes such that divides . Let be the semidirect product of the group of order by the subgroup of order in its automorphism group. is a group of order . Its lattice has size , including the trivial subgroup, whole group, and intermediate mutually incomparable subgroups, one of order and of order .
Let be the subgroup of order and be any subgroup of order . The map from to itself interchanging and is a lattice automorphism, and it interchanges the two subgroups. Also, and satisfy the stated conditions, completing the proof.
The smallest example is obtained by setting , giving:
For more on the subgroup structure, see subgroup structure of symmetric group:S3.
The lattice of subgroups is also shown below: