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

A subgroup H of a group G is termed divisibility-closed or divisibility-invariant if it satisfies the following equivalent conditions:

  1. For every prime p such that G is p-divisible (i.e., for every g \in G, there exists x \in G such that x^p = g), H is also p-divisible.
  2. For every natural number n such that G is n-divisible (i.e., for every g \in G, there exists x \in G such that x^n = g), H is also n-divisible.

Note that we do not require that all the n^{th} roots in G of an element of H must be in H.

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
transitive subgroup property Yes divisibility-closedness is transitive If H \le K \le G are groups such that H is divisibility-closed in K and K is divisibility-closed in G, then H is divisibility-closed in G.
trim subgroup property Yes In any group, the whole group and the trivial subgroup are divisibility-closed.
finite-intersection-closed subgroup property No divisibility-closedness is not finite-intersection-closed It is possible to have a group G and divisibility-closed subgroups H,K of G such that the intersection of subgroups H \cap K is not divisibility-closed in G.
finite-join-closed subgroup property No divisibility-closedness is not finite-join-closed It is possible to have a group G and divisibility-closed subgroups H,K of G such that the join of subgroups \langle H, K \rangle is not divisibility-closed in G.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
direct factor normal subgroup with normal complement (via retract)
retract has a normal complement Endomorphism image, Local divisibility-closed subgroup, Verbally closed subgroup|FULL LIST, MORE INFO
subgroup of finite group Completely divisibility-closed subgroup|FULL LIST, MORE INFO
local divisibility-closed subgroup if a particular element of the subgroup has a n^{th} root in the whole group, the element has a n^{th} root in the subgroup. |FULL LIST, MORE INFO
completely divisibility-closed subgroup divisibility-closed, and all relevant roots lie inside the subgroup. |FULL LIST, MORE INFO
kernel of a bihomomorphism kernel of a bihomomorphism (via completely divisibility-closed) Completely divisibility-closed subgroup|FULL LIST, MORE INFO
kernel of a multihomomorphism (via completely divisibility-closed) Completely divisibility-closed subgroup|FULL LIST, MORE INFO
subgroup of finite index (via completely divisibility-closed) |FULL LIST, MORE INFO
pure subgroup local divisibility-closed subgroup of abelian group (via local divisibility-closed) any example in a non-abelian group |FULL LIST, MORE INFO
verbal subgroup of abelian group verbal subgroup of abelian group implies divisibility-closed Divisibility-closed subgroup of abelian group|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
powering-invariant subgroup powering-invariant not implies divisibility-closed

Satisfaction by subgroup-defining functions

Subgroup-defining function Always divisibility-closed? Proof Stronger/weaker properties satisfied If the group property is restricted
center No center not is divisibility-closed weaker: powering-invariant subgroup (see center is powering-invariant and center is local powering-invariant) in nilpotent group, center is a completely divisibility-closed subgroup (follows from upper central series members are completely divisibility-closed in nilpotent group)
members of the upper central series No based on failure for center weaker: powering-invariant subgroup (see upper central series members are powering-invariant, upper central series members are quotient-powering-invariant) in nilpotent group, they are divisibility-closed subgroups, and in fact are completely divisibility-closed subgroups (follows from upper central series members are completely divisibility-closed in nilpotent group)
derived subgroup No derived subgroup not is divisibility-closed in nilpotent group, yes (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 No based on failure for derived subgroup in nilpotent group, they are divisibility-closed (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 No based on failure for derived subgroup in nilpotent group, they are divisibility-closed subgroups. This follows from the result for the derived subgroup. See derived series members are divisibility-closed in nilpotent group.
socle No Not true even in abelian groups, see monolith not is divisibility-closed in abelian group. However, the weaker property of being a powering-invariant subgroup is true in solvable groups: socle is powering-invariant in solvable group