Local powering-invariant subgroup

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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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)
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

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 $n = 1$.

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.