Finite direct power-closed characteristic subgroup

Definition

Suppose $H$ is a subgroup of a group $G$. We say that $H$ is finite direct power-closed characteristic in $G$ if the following holds for every natural number $n$: in the group $G^n$ which is defined as the external direct product of $n$ copies of $G$, the corresponding subgroup $H^n$ is a characteristic subgroup.

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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 in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property finite direct power-closed characteristic subgroup.

Group partSubgroup partQuotient part
A3 in S3Symmetric group:S3Cyclic group:Z3Cyclic group:Z2
A4 in S4Symmetric group:S4Alternating group:A4Cyclic group:Z2
Center of M16M16Cyclic group:Z4Klein four-group
Center of dihedral group:D16Dihedral group:D16Cyclic group:Z2Dihedral group:D8
Center of dihedral group:D8Dihedral group:D8Cyclic group:Z2Klein four-group
Center of direct product of D8 and Z2Direct product of D8 and Z2Klein four-groupKlein four-group
Center of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Klein four-groupKlein four-group
Center of quaternion groupQuaternion groupCyclic group:Z2Klein four-group
Center of semidihedral group:SD16Semidihedral group:SD16Cyclic group:Z2Dihedral group:D8
Center of special linear group:SL(2,3)Special linear group:SL(2,3)Cyclic group:Z2Alternating group:A4
Center of special linear group:SL(2,5)Special linear group:SL(2,5)Cyclic group:Z2Alternating group:A5
Center of unitriangular matrix group:UT(3,p)Unitriangular matrix group:UT(3,p)Group of prime orderElementary abelian group of prime-square order
Cyclic maximal subgroup of dihedral group:D8Dihedral group:D8Cyclic group:Z4Cyclic group:Z2
D8 in SD16Semidihedral group:SD16Dihedral group:D8Cyclic group:Z2
Derived subgroup of M16M16Cyclic group:Z2Direct product of Z4 and Z2
Derived subgroup of dihedral group:D16Dihedral group:D16Cyclic group:Z4Klein four-group
Derived subgroup of nontrivial semidirect product of Z4 and Z4Nontrivial semidirect product of Z4 and Z4Cyclic group:Z2Direct product of Z4 and Z2
Direct product of Z4 and Z2 in M16M16Direct product of Z4 and Z2Cyclic group:Z2
First agemo subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Klein four-group
First omega subgroup of direct product of Z4 and Z2Direct product of Z4 and Z2Klein four-groupCyclic group:Z2
Klein four-subgroup of M16M16Klein four-groupCyclic group:Z4
Klein four-subgroup of alternating group:A4Alternating group:A4Klein four-groupCyclic group:Z3
Normal Klein four-subgroup of symmetric group:S4Symmetric group:S4Klein four-groupSymmetric group:S3
SL(2,3) in GL(2,3)General linear group:GL(2,3)Special linear group:SL(2,3)Cyclic group:Z2

Below are some examples of a proper nontrivial subgroup that does not satisfy the property finite direct power-closed characteristic subgroup.

Group partSubgroup partQuotient part
Cyclic maximal subgroups of quaternion groupQuaternion groupCyclic group:Z4Cyclic group:Z2
D8 in D16Dihedral group:D16Dihedral group:D8Cyclic group:Z2
Diagonally embedded Z4 in direct product of Z8 and Z2Direct product of Z8 and Z2Cyclic group:Z4Cyclic group:Z4
Group of integers in group of rational numbersGroup of rational numbersGroup of integersGroup of rational numbers modulo integers
Klein four-subgroups of dihedral group:D8Dihedral group:D8Klein four-groupCyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z2Cyclic group:Z4
S2 in S3Symmetric group:S3Cyclic group:Z2
Z2 in V4Klein four-groupCyclic group:Z2Cyclic group:Z2
Z4 in direct product of Z4 and Z2Direct product of Z4 and Z2Cyclic group:Z4Cyclic group:Z2

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes finite direct power-closed characteristic is transitive If $H \le K \le G$, with $H$ a finite direct power-closed characteristic subgroup of $K$ and $K$ a finite direct power-closed characteristic subgroup of $G$, then $H$ is a finite direct power-closed characteristic subgroup of $G$.
quotient-transitive subgroup property Yes finite direct power-closed characteristic is quotient-transitive If $H \le K \le G$, with $H$ a finite direct power-closed characteristic subgroup of $G$ and $K/H$ a finite direct power-closed characteristic subgroup of $G/H$, then $K$ is a finite direct power-closed characteristic subgroup of $G$.
strongly intersection-closed subgroup property Yes finite direct power-closed characteristic is strongly intersection-closed If $H_i, i \in I$ are all finite direct power-closed characteristic subgroups of $G$, so is the intersection $\bigcap_i H_i$.
centralizer-closed subgroup property Yes finite direct power-closed characteristic is centralizer-closed If $H \le G$ is a finite direct power-closed characteristic subgroup, so is the centralizer $C_G(H)$.
commutator-closed subgroup property Yes finite direct power-closed characteristic is commutator-closed If $H, K \le G$ are both finite direct power-closed characteristic, then so is the commutator $[H,K]$.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
direct power-closed characteristic subgroup analogous condition, but we allow infinite unrestricted direct powers |FULL LIST, MORE INFO
restricted direct power-closed characteristic subgroup analogous condition, but we allow infinite restricted direct powers |FULL LIST, MORE INFO
fully invariant subgroup invariant under all endomorphisms combine full invariance is finite direct power-closed and fully invariant implies characteristic finite direct power-closed characteristic not implies fully invariant Normality-preserving endomorphism-invariant subgroup|FULL LIST, MORE INFO
normality-preserving endomorphism-invariant subgroup invariant under all normality-preserving endomorphisms normality-preserving endomorphism-invariant implies finite direct power-closed characteristic |FULL LIST, MORE INFO
bound-word subgroup given by a system of conditions with quantifiers on all other variables bound-word implies finite direct power-closed characteristic  ? |FULL LIST, MORE INFO
verbal subgroup Fully invariant subgroup|FULL LIST, MORE INFO
marginal subgroup (via bound-word) (via (bound-word) Direct power-closed characteristic subgroup, Sub-weakly marginal subgroup, Submarginal subgroup|FULL LIST, MORE INFO

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