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]
A subgroup of a group is termed powering-invariant if it satisfies the following equivalent conditions:
|No.||Shorthand||A subgroup of a group is powering-invariant in if ...|
|1||powered over primes the group is powered over||for every prime such that is powered over (i.e., it is uniquely -divisible), is also powered over .|
|2||invariant under well-defined root maps||for every prime such that for all , there exists unique such that , if we denote , we have that the subgroup is invariant under the function .|
|3||invariant under well-defined root maps||for every natural number such that for all , there exists unique such that , if we denote , we have that the subgroup is invariant under the function .|
|4||invariant under well-defined rational power maps||for every rational number (in reduced form) such that for all , there is a unique satisfying if we denote , we have that the subgroup is invariant under the function .|
|5||invariant under well-defined rational power maps (slight variant)||for every rational number (in reduced form) such that every element has a unique root, if we denote by the unique root of , we have that the subgroup is invariant under the function .|
|Metaproperty name||Satisfied?||Proof||Statement with symbols|
|transitive subgroup property||Yes||powering-invariance is transitive||If are such that is powering-invariant in and is powering-invariant in , then is powering-invariant in .|
|strongly intersection-closed subgroup property||Yes||powering-invariance is strongly intersection-closed||If is a (finite or infinite, possibly empty) collection of powering-invariant subgroups of a group , then the intersection is also powering-invariant.|
|intermediate subgroup condition||No||powering-invariance does not satisfy intermediate subgroup condition||It is possible to have groups such that is powering-invariant in but not in .|
|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 and powering-invariant subgroups and of such that is not powering-invariant in .|
|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 such that is powering-invariant and normal in and is powering-invariant in but is not powering-invariant in .|
|centralizer-closed subgroup property||Yes||powering-invariance is centralizer-closed (follows from c-closed implies powering-invariant)||If is a group and is a powering-invariant subgroup of , then the centralizer is also powering-invariant. In fact, is powering-invariant even if 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 and powering-invariant subgroups of such that the commutator is not powering-invariant.|
|union-closed subgroup property||Yes||powering-invariance is union-closed||If are all powering-invariant subgroups of a group , and their set-theoretic union is a subgroup , then is also a powering-invariant subgroup of .|
|lower central series condition||No||powering-invariance does not satisfy lower central series condition||It is possible to choose a group , a powering-invariant subgroup of , and a positive integer such that is not a powering-invariant subgroup of .|
Relation with other properties
For more on the background, see subgroup-quotient duality for groups.
The dual property to this is quotient-powering-invariant subgroup. A subgroup of a group is termed quotient-powering-invariant in if is a normal subgroup of and for every prime number such that is -powered, is also -powered.
|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
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 Function
Here, "rational power map" refers to a map of the form where is a rational number for which the map is well-defined (see definitions no. (4) or (5)).
The expression can be right-tightened to give a balanced subgroup property:
Rational power map Rational power map
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.