Local 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]
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)
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 Powering-invariant subgroup|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 Powering-invariant subgroup|FULL LIST, MORE INFO

Facts