# Finite direct power-closed characteristic subgroup

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## 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