Powering-invariant subgroup

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Definition

A subgroup $H$ of a group $G$ is termed powering-invariant if it satisfies the following equivalent conditions:

No.	Shorthand	A subgroup $H$ of a group $G$ is powering-invariant in $G$ if ...
1	powered over primes the group is powered over	for every prime $p$ such that $G$ is powered over $p$ (i.e., it is uniquely $p$-divisible), $H$ is also powered over $p$.
2	invariant under well-defined $p^{th}$ root maps	for every prime $p$ such that for all $g\in G$, there exists unique $x$ such that $x^{p}=g$, if we denote $x=g^{1/p}$, we have that the subgroup $H$ is invariant under the function $g\mapsto g^{1/p}$.
3	invariant under well-defined $n^{th}$ root maps	for every natural number $n$ such that for all $g\in G$, there exists unique $x$ such that $x^{n}=g$, if we denote $x=g^{1/n}$, we have that the subgroup $H$ is invariant under the function $g\mapsto g^{1/n}$.
4	invariant under well-defined rational power maps	for every rational number $r=a/b$ (in reduced form) such that for all $g\in G$, there is a unique $x\in G$ satisfying $g^{a}=x^{b}$ if we denote $x=g^{r}$, we have that the subgroup $H$ is invariant under the function $g\mapsto g^{r}$.
5	invariant under well-defined rational power maps (slight variant)	for every rational number $r=a/b$ (in reduced form) such that every element has a unique $b^{th}$ root, if we denote by $g^{r}$ the unique $b^{th}$ root of $g^{a}$, we have that the subgroup $H$ is invariant under the function $g\mapsto g^{r}$.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	powering-invariance is transitive	If $H\leq K\leq G$ are such that $H$ is powering-invariant in $K$ and $K$ is powering-invariant in $G$, then $H$ is powering-invariant in $G$.
strongly intersection-closed subgroup property	Yes	powering-invariance is strongly intersection-closed	If $H_{i},i\in I$ is a (finite or infinite, possibly empty) collection of powering-invariant subgroups of a group $G$, then the intersection $\bigcap _{i\in I}H_{i}$ is also powering-invariant.
intermediate subgroup condition	No	powering-invariance does not satisfy intermediate subgroup condition	It is possible to have groups $H\leq K\leq G$ such that $H$ is powering-invariant in $G$ but not in $K$.
finite-join-closed subgroup property	No	powering-invariance is not finite-join-closed (note, however, that powering-invariance is strongly join-closed in nilpotent group)	It is possible to have a group $G$ and powering-invariant subgroups $H$ and $K$ of $G$ such that $\langle H,K\rangle$ is not powering-invariant in $G$.
quotient-transitive subgroup property	No	powering-invariance is not quotient-transitive, but see powering-invariant over quotient-powering-invariant implies powering-invariant	It is possible to have groups $H\leq K\leq G$ such that $H$ is powering-invariant and normal in $G$ and $K/H$ is powering-invariant in $G/H$ but $K$ is not powering-invariant in $G$.
centralizer-closed subgroup property	Yes	powering-invariance is centralizer-closed (follows from c-closed implies powering-invariant)	If $G$ is a group and $H$ is a powering-invariant subgroup of $G$, then the centralizer $C_{G}(H)$ is also powering-invariant. In fact, $C_{G}(H)$ is powering-invariant even if $H$ is not.
commutator-closed subgroup property	No	powering-invariance is not commutator-closed (note, however, that powering-invariance is commutator-closed in nilpotent group)	It is possible to have a group $G$ and powering-invariant subgroups $H,K$ of $G$ such that the commutator $[H,K]$ is not powering-invariant.
union-closed subgroup property	Yes	powering-invariance is union-closed	If $H_{i},i\in I$ are all powering-invariant subgroups of a group $G$, and their set-theoretic union $\bigcup _{i\in I}H_{i}$ is a subgroup $H$, then $H$ is also a powering-invariant subgroup of $G$.
lower central series condition	No	powering-invariance does not satisfy lower central series condition	It is possible to choose a group $G$, a powering-invariant subgroup $H$ of $G$, and a positive integer $k$ such that $\gamma _{k}(H)$ is not a powering-invariant subgroup of $\gamma _{k}(G)$.

Relation with other properties

Dual property

For more on the background, see subgroup-quotient duality for groups.

The dual property to this is quotient-powering-invariant subgroup. A subgroup $H$ of a group $G$ is termed quotient-powering-invariant in $G$ if $H$ is a normal subgroup of $G$ and for every prime number $p$ such that $G$ is $p$-powered, $G/H$ is also $p$-powered.

The correspondence we use is subgroup $\leftrightarrow$ quotient group.

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
local powering-invariant subgroup	for any $h\in H$ and any natural number $n$ such that there is a unique $x\in G$ satisfying $x^{n}=h$, we must have $x\in H$.		powering-invariant not implies local powering-invariant	|FULL LIST, MORE INFO
characteristic subgroup of abelian group	characteristic subgroup and the whole group is an abelian group	characteristic subgroup of abelian group implies powering-invariant		|FULL LIST, MORE INFO
finite subgroup		finite implies powering-invariant		|FULL LIST, MORE INFO
periodic subgroup				|FULL LIST, MORE INFO
subgroup of finite index		finite index implies powering-invariant		|FULL LIST, MORE INFO
normal subgroup of finite index		(via subgroup of finite index, also via quotient-powering-invariant subgroup)	(via subgroup of finite index)	|FULL LIST, MORE INFO
quotient-powering-invariant subgroup	normal subgroup such that if the whole group is powered over a prime, so is the quotient group.	quotient-powering-invariant implies powering-invariant	powering-invariant not implies quotient-powering-invariant	|FULL LIST, MORE INFO
divisibility-closed subgroup			powering-invariant not implies divisibility-closed	|FULL LIST, MORE INFO
local divisibility-closed subgroup		(via divisibility-closed)	(via divisibility-closed)	|FULL LIST, MORE INFO
retract	has a normal complement	(via local divisibility-closed)	(via local divisibility-closed)	|FULL LIST, MORE INFO
endomorphism image	image of the whole group under an endomorphism	(via divisibility-closed)	any subgroup of a finite group that is not an endomorphic image (e.g., any proper nontrivial subgroup of a finite simple non-abelian group)	|FULL LIST, MORE INFO
endomorphism kernel	kernel of an endomorphism	(via quotient-powering-invariant)	(via quotient-powering-invariant)	|FULL LIST, MORE INFO
complemented normal subgroup	normal subgroup with a permutable complement, i.e., part of an internal semidirect product	(via endomorphism kernel)	(via endomorphism kernel)	|FULL LIST, MORE INFO
direct factor	factor in an internal direct product	(via complemented normal, also via retract)	(via complemented normal, also via retract)	|FULL LIST, MORE INFO
fixed-point subgroup of a subgroup of the automorphism group	fixed-point subgroup of a subgroup of the automorphism group	(via local powering-invariant)	(via local powering-invariant)	|FULL LIST, MORE INFO
c-closed subgroup	centralizer of some subset of the group	(via fixed-point subgroup of a subgroup of the automorphism group)	(via fixed-point subgroup of a subgroup of the automorphism group)	|FULL LIST, MORE INFO

Incomparable properties

Property	Meaning	Proof that it does not imply being powering-invariant	Properties for the ambient group for which it does imply being powering-invariant	Proof that being powering-invariant does not imply the property	Property conjunction
normal subgroup	invariant under all inner automorphisms	see examples for characteristic or central subgroup below.	finite group or periodic group	any finite non-normal subgroup	powering-invariant normal subgroup
characteristic subgroup	invariant under all automorphisms	characteristic not implies powering-invariant	abelian group (plus above cases for normal subgroup) and perhaps nilpotent group	any finite non-characteristic subgroup	powering-invariant characteristic subgroup
central subgroup	contained in the center	follows from subgroup of abelian group not implies powering-invariant	(all cases for normal subgroup)	any non-abelian group as a subgroup of itself	powering-invariant central subgroup

Satisfaction by subgroup-defining functions

Subgroup-defining function	Always powering-invariant?	Proof	Stronger/weaker properties satisfied	If the group property is restricted
center	Yes	center is powering-invariant	stronger: local powering-invariant subgroup (see center is local powering-invariant)	in nilpotent group: completely divisibility-closed subgroup (follows from upper central series members are completely divisibility-closed in nilpotent group), intermediately local powering-invariant subgroup (follows from upper central series members are intermediately local powering-invariant in nilpotent group)
members of the upper central series	Yes	upper central series members are powering-invariant	quotient-powering-invariant subgroup (see upper central series members are quotient-powering-invariant)	in nilpotent group, completely divisibility-closed subgroup (follows from upper central series members are completely divisibility-closed in nilpotent group), intermediately local powering-invariant subgroup (follows from upper central series members are intermediately local powering-invariant in nilpotent group)
derived subgroup	Unclear, probably no	pending		in nilpotent group, it is powering-invariant and in fact, divisibility-closed subgroup (see derived subgroup is divisibility-closed in nilpotent group, follows from equivalence of definitions of nilpotent group that is divisible for a set of primes)
members of the lower central series	Unclear, probably no	pending		in nilpotent group, they are powering-invariant and in fact are divisibility-closed subgroups (see lower central series members are divisibility-closed in nilpotent group, follows from equivalence of definitions of nilpotent group that is divisible for a set of primes)
members of the derived series	Unclear, probably no	pending		in nilpotent group, they are powering-invariant and in fact are divisibility-closed subgroups. This follows from the result for the derived subgroup.
socle	Unclear, probably no	pending		in solvable group: see socle is powering-invariant in solvable group

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

A function restriction expression for the property of being a powering-invariant subgroup is as follows:

Rational power map $\to$ Function

Here, "rational power map" refers to a map of the form $x\mapsto x^{r}$ where $r$ is a rational number for which the map is well-defined (see definitions no. (4) or (5)).

This shows that the property is an invariance property, which immediately implies that it is, among other things, strongly intersection-closed.

The expression can be right-tightened to give a balanced subgroup property:

Rational power map $\to$ Rational power map

It being a balanced subgroup property immediately shows that it's a transitive subgroup property (since balanced implies transitive).

Above, we could replace "rational power map" throughout to a subset of rational power maps, such as prime root maps or integer root maps (see definitions (2) or (3)). We chose "rational power map" because the well-defined rational power maps on a group form a subquotient of the multiplicative rationals and are the largest such permitted subquotient.