Quotient-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]
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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and normal subgroup satisfying the subgroup-to-quotient powering-invariance implication
View other subgroup property conjunctions | view all subgroup properties

Definition

A normal subgroup H of a group G is termed a quotient-powering-invariant subgroup if, for any prime number p such that G is a powered for p, the quotient group G/H is also powered for p.

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
quotient-transitive subgroup property Yes quotient-powering-invariance is quotient-transitive If H \le K \le G are such that H is quotient-powering-invariant in G and K/H is quotient-powering-invariant in G/H, then K is quotient-powering-invariant in G.
union-closed subgroup property Yes quotient-powering-invariance is union-closed If H_i, i \in I are all quotient-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 quotient-powering-invariant subgroup of G.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
normal subgroup of finite group the whole group is finite
normal subgroup of periodic group every element in the whole group has finite order
normal subgroup of finite index the quotient group is finite normal of finite index implies quotient-powering-invariant
finite normal subgroup the normal subgroup is finite finite normal implies quotient-powering-invariant
endomorphism kernel normal subgroup that is the kernel of an endomorphism endomorphism kernel implies quotient-powering-invariant any normal subgroup of a finite group that is not an endomorphism kernel works. |FULL LIST, MORE INFO
complemented normal subgroup normal subgroup with a (possibly non-normal) complement (via endomorphism kernel, see also direct proof) Endomorphism kernel|FULL LIST, MORE INFO
direct factor normal subgroup with normal complement (via complemented normal) (via complemented normal) Endomorphism kernel|FULL LIST, MORE INFO
characteristic subgroup of abelian group characteristic subgroup and the whole group is an abelian group characteristic subgroup of abelian group is quotient-powering-invariant Characteristic subgroup of center, Quotient-powering-invariant characteristic subgroup|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
powering-invariant normal subgroup quotient-powering-invariant implies powering-invariant powering-invariant and normal not implies quotient-powering-invariant |FULL LIST, MORE INFO
powering-invariant subgroup quotient-powering-invariant implies powering-invariant (via powering-invariant normal) Powering-invariant normal subgroup|FULL LIST, MORE INFO
normal subgroup satisfying the subgroup-to-quotient powering-invariance implication |FULL LIST, MORE INFO
normal subgroup (by definition) (via powering-invariant normal) Powering-invariant normal subgroup|FULL LIST, MORE INFO

Properties whose conjunction with powering-invariance implies quotient-powering-invariance

The relevant subgroup property is normal subgroup satisfying the subgroup-to-quotient powering-invariance implication. In fact, the conjunction of this with powering-invariant subgroup precisely gives quotient-powering-invariant subgroup.

This property is implied both by being a central subgroup and by being a normal subgroup contained in the hypercenter.