Quotient-torsion-freeness-closed subgroup

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Quotient-torsion-freeness-closed subgroup, all facts related to Quotient-torsion-freeness-closed subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

Definition

Suppose $G$ is a group and $H$ is a normal subgroup of $G$. We say that $H$ is a quotient-torsion-freeness-closed subgroup of $G$ if $H$ is a normal subgroup of $G$ and for any prime number $p$ such that $G$ is $p$-torsion-free, the quotient group $G/H$ is also $p$-torsion-free.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
quotient-local powering-invariant subgroup	if an element of the group has a unique $n^{th}$ root, the same is true of its image in the quotient group.			|FULL LIST, MORE INFO