Powering-invariant subgroup

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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]

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 \le K \le 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 \le K \le 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 \le K \le 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 Characteristic subgroup of center, Intermediately powering-invariant subgroup, Powering-invariant characteristic subgroup, Powering-invariant normal subgroup, Powering-invariant subgroup of abelian group, Quotient-powering-invariant characteristic subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
finite subgroup finite implies powering-invariant Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup, LCS-powering-invariant subgroup, Periodic subgroup|FULL LIST, MORE INFO
periodic subgroup Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup|FULL LIST, MORE INFO
subgroup of finite index finite index implies powering-invariant Divisibility-closed subgroup, Intermediately powering-invariant subgroup|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) Intermediately powering-invariant subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|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 Powering-invariant normal subgroup|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) Divisibility-closed subgroup, Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup|FULL LIST, MORE INFO
retract has a normal complement (via local divisibility-closed) (via local divisibility-closed) Divisibility-closed subgroup, Endomorphism image, Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup, Local divisibility-closed subgroup, Verbally closed subgroup|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) Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|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) Endomorphism kernel, Intermediately powering-invariant subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup|FULL LIST, MORE INFO
direct factor factor in an internal direct product (via complemented normal, also via retract) (via complemented normal, also via retract) Divisibility-closed subgroup, Endomorphism image, Endomorphism kernel, Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup, Local divisibility-closed subgroup, Powering-invariant normal subgroup, Quotient-powering-invariant subgroup, Retract, Verbally closed subgroup|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) 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.