Local divisibility-closed subgroup

From Groupprops
Jump to: navigation, search
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

Suppose G is a group and H is a subgroup of G. We say that H is local divisibility-closed or local divisibility-invariant in G if the following holds: for any h \in H and any natural number n such that the equation x^n = h has a solution for x in G, the equation x^n = h has a solution for x \in H.

Relation with other properties

Conjunction with group properties for ambient group

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal Sylow subgroup
normal Hall subgroup
retract Verbally closed subgroup|FULL LIST, MORE INFO
direct factor Retract, Verbally closed subgroup|FULL LIST, MORE INFO
verbally closed subgroup |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
local powering-invariant subgroup Intermediately local powering-invariant subgroup|FULL LIST, MORE INFO
divisibility-closed subgroup divisibility-closed not implies local divisibility-closed |FULL LIST, MORE INFO
powering-invariant subgroup Divisibility-closed subgroup, Intermediately local powering-invariant subgroup, Intermediately powering-invariant subgroup, Local powering-invariant subgroup|FULL LIST, MORE INFO

Incomparable properties

Facts