Finite direct power-closed characteristic subgroup
From Groupprops
Definition
Suppose is a subgroup of a group
. We say that
is finite direct power-closed characteristic in
if the following holds for every natural number
: in the group
which is defined as the external direct product of
copies of
, the corresponding subgroup
is a characteristic subgroup.
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is a variation of characteristic subgroup|Find other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup
Examples
Extreme examples
- The trivial subgroup in any group is a finite direct power-closed characteristic subgroup.
- Every group is a finite direct power-closed characteristic subgroup of itself.
Examples of subgroup-defining functions
- Center is finite direct power-closed characteristic: The center of a group always has this property. More generally, any bound-word subgroup, and in particular any marginal subgroup, is finite direct power-closed characteristic. Thus, all members of the finite part of the upper central series are finite direct power-closed characteristic.
- The derived subgroup, and more generally any verbal subgroup or even any fully invariant subgroup, is finite direct power-closed characteristic. Hence, all members of the finite part of the lower central series as well as of the derived series are finite direct power-closed characteristic.
Examples in small finite groups
Below are some examples of a proper nontrivial subgroup that satisfy the property finite direct power-closed characteristic subgroup.
Below are some examples of a proper nontrivial subgroup that does not satisfy the property finite direct power-closed characteristic subgroup.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | Yes | finite direct power-closed characteristic is transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
quotient-transitive subgroup property | Yes | finite direct power-closed characteristic is quotient-transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
strongly intersection-closed subgroup property | Yes | finite direct power-closed characteristic is strongly intersection-closed | If ![]() ![]() ![]() |
centralizer-closed subgroup property | Yes | finite direct power-closed characteristic is centralizer-closed | If ![]() ![]() |
commutator-closed subgroup property | Yes | finite direct power-closed characteristic is commutator-closed | If ![]() ![]() |
Relation with other properties
Stronger properties
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
characteristic subgroup | invariant under all automorphisms | (by definition) | characteristicity is not finite direct power-closed | |FULL LIST, MORE INFO |
normal subgroup | invariant under all inner automorphisms | (via characteristic) | (via characteristic) | Characteristic subgroup|FULL LIST, MORE INFO |