Local powering-invariant subgroup

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Definition

A subgroup $H$ of a group $G$ is termed a local powering-invariant subgroup if the following hold:

1. Whenever $h \in H$ and $n \in \mathbb{N}$ are such that there is a unique $x \in G$ such that $x^n = h$, we must have $x \in H$.
2. Whenever $h \in H$ and $p$ is a prime number such that there is a unique $x \in G$ such that $x^p = h$, we must have $x \in H$.

Equivalence of definitions

Further information: equivalence of definitions of local powering-invariant subgroup

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	local powering-invariance is transitive	Suppose $H \le K \le G$ are groups such that $H$ is local powering-invariant in $K$ and $K$ is local powering-invariant in $G$. Then $H$ is local powering-invariant in $G$.
strongly intersection-closed subgroup property	Yes	local powering-invariance is strongly intersection-closed	Suppose $H_i, i \in I$ are all local powering-invariant subgroups of a group $G$. Then, the intersection of subgroups $\bigcap_{i \in I} H_i$ is also a local powering-invariant subgroup of $G$.
intermediate subgroup condition	No	local powering-invariance does not satisfy intermediate subgroup condition	It is possible to have groups $H \le K \le G$ such that $H$ is local powering-invariant in $G$ but not in $K$.
finite-join-closed subgroup property	No	local powering-invariance is not finite-join-closed	It is possible to have a group $G$ and local powering-invariant subgroups $H$ and $K$ of $G$ such that the join of subgroups $\langle H, K \rangle$ is not local powering-invariant.
centralizer-closed subgroup property	Yes	follows from c-closed implies local powering-invariant	Suppose $G$ is a group and $H$ is a local powering-invariant subgroup of $G$. Then, the centralizer $C_G(H)$ is also a local powering-invariant subgroup of $G$. In fact, we do not even need $H$ to be local powering-invariant -- the assumption is redundant.
commutator-closed subgroup property	No	local powering-invariance is not commutator-closed	It is possible to have a group $G$ and subgroups $H,K$ of $G$ such that both $H$ and $K$ are local powering-invariant in $G$ but the commutator $[H,K]$ is not.
image condition	No	local powering-invariance does not satisfy image condition	It is possible to have a group $G$, a local powering-invariant subgroup $H$ of $G$, and a surjective homomorphism $\varphi:G \to K$ such that $\varphi(H)$ is not local powering-invariant in $K$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite subgroup		finite implies local powering-invariant
periodic subgroup
fixed-point subgroup of a subgroup of the automorphism group		fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
c-closed subgroup		(via fixed-point subgroup of a subgroup of the automorphism group)
local divisibility-closed subgroup				|FULL LIST, MORE INFO
retract	has a normal complement	(via local divisibility-closed)	(via local divisibility-closed)	Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Verbally closed subgroup|FULL LIST, MORE INFO
direct factor	normal subgroup with normal complement	(via retract)	(via retract)	Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Retract, Verbally closed subgroup|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
powering-invariant subgroup

Incomparable properties

Property	Meaning	Proof that it does not imply being local powering-invariant	Proof that being local powering-invariant does not imply it	Properties stronger than both	Properties weaker than both
subgroup of finite index	has finite index in the whole group	finite index not implies local powering-invariant	any direct factor where the other direct factor is infinite.	|FULL LIST, MORE INFO	|FULL LIST, MORE INFO
complemented normal subgroup	normal subgroup with a permutable complement	complemented normal not implies local powering-invariant	any non-normal subgroup of a finite group will do.	Direct factor|FULL LIST, MORE INFO	|FULL LIST, MORE INFO

Facts

• Center is local powering-invariant
• Derived subgroup not is local powering-invariant

Formalisms

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

The property of being a local powering-invariant subgroup can be expressed as an invariance property with respect to local powerings:

Local powering $\to$ Function

Here, we can define a "local powering" as a function that takes every element to its unique $n^{th}$ root for some $n$ that could vary with the element. Note that any element that lacks a nontrivial unique root can just be sent to itself, using $n = 1$.

Since invariance implies strongly intersection-closed, this shows that local powering-invariance is strongly intersection-closed.

The property can be right tightened to a balanced subgroup property:

Local powering $\to$ Local powering

Since balanced implies transitive, this shows that local powering-invariance is transitive.