Completely divisibility-closed 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]
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Definition

Suppose G is a group. A subgroup H of G is termed completely divisibility-closed if the following holds: for any prime number p such that G is p-divisible, and any g \in H, all p^{th} roots of g in G lie inside H.

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes complete divisibility-closedness is transitive If H \le K \le G are groups such that H is completely divisibility-closed in K and K is completely divisibility-closed in G, then H is completely divisibility-closed in G.
strongly intersection-closed subgroup property Yes complete divisibility-closedness is strongly intersection-closed If H_i, i \in I are completely divisibility-closed subgroups of G, so is \bigcap_{i \in I} H_i.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
completely divisibility-closed normal subgroup completely divisibility-closed and normal; equivalently, the quotient is torsion-free for any prime for which the whole group is divisible. |FULL LIST, MORE INFO
kernel of a bihomomorphism kernel of a bihomomorphism implies completely divisibility-closed
intersection of kernels of bihomomorphisms intersection of kernels of bihomomorphisms implies completely divisibility-closed |FULL LIST, MORE INFO
kernel of a multihomomorphism kernel of a multihomomorphism implies completely divisibility-closed Intersection of kernels of bihomomorphisms|FULL LIST, MORE INFO
subgroup of finite group
subgroup of periodic group

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
divisibility-closed subgroup |FULL LIST, MORE INFO
powering-invariant subgroup Divisibility-closed subgroup|FULL LIST, MORE INFO