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 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.

	Group part	Subgroup part	Quotient part
Center of M16	M16	Cyclic group:Z4	Klein four-group
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group
Center of direct product of D8 and Z2	Direct product of D8 and Z2	Klein four-group	Klein four-group
Cyclic maximal subgroup of dihedral group:D8	Dihedral group:D8	Cyclic group:Z4	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 part	Subgroup part	Quotient part
Cyclic maximal subgroups of quaternion group	Quaternion group	Cyclic group:Z4	Cyclic group:Z2
D8 in D16	Dihedral group:D16	Dihedral group:D8	Cyclic group:Z2
Diagonally embedded Z4 in direct product of Z8 and Z2	Direct product of Z8 and Z2	Cyclic group:Z4	Cyclic group:Z4
Group of integers in group of rational numbers	Group of rational numbers	Group of integers	Group of rational numbers modulo integers
Klein four-subgroups of dihedral group:D8	Dihedral group:D8	Klein four-group	Cyclic group:Z2
Non-characteristic order two subgroups of direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z2	Cyclic group:Z4
S2 in S3	Symmetric group:S3	Cyclic group:Z2
Z2 in V4	Klein four-group	Cyclic group:Z2	Cyclic group:Z2
Z4 in direct product of Z4 and Z2	Direct product of Z4 and Z2	Cyclic group:Z4	Cyclic 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				|FULL LIST, MORE INFO
marginal subgroup		(via bound-word)	(via (bound-word)	Direct power-closed characteristic subgroup, Sub-weakly marginal subgroup, Submarginal subgroup, Weakly marginal 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