Statement
It is possible to have a group
, a lattice automorphism
of the lattice of subgroups, and subgroups
of
with
, such that:
Property-theoretic implication
No subgroup property of any of the following kinds is a conditionally lattice-determined subgroup property:
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This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about subnormal subgroup|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., normal subgroup) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about normal subgroup|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., characteristic subgroup) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about characteristic subgroup|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., fully invariant subgroup) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about fully invariant subgroup|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., normal Hall subgroup) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about normal Hall subgroup|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., normal Sylow subgroup) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about normal Sylow subgroup|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., complemented normal subgroup) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about complemented normal subgroup|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., Sylow retract) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about Sylow retract|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall retract) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about Hall retract|Get more facts about conditionally lattice-determined subgroup property|
This article gives the statement, and possibly proof, of a subgroup property (i.e., retract) not satisfying a subgroup metaproperty (i.e., conditionally lattice-determined subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about retract|Get more facts about conditionally lattice-determined subgroup property|
Related facts
Proof
General example
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.
Smallest example
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: