Varietal group property: Difference between revisions

Latest revision as of 22:05, 12 January 2024

This article defines a group metaproperty: a property that can be evaluated to true/false for any group property
View a complete list of group metaproperties

Definition

A group property $α$ is termed varietal if it satisfies the following three conditions:

It is a subgroup-closed group property, i.e., whenever $G$ is a group satisfying $α$ and $H$ is a subgroup of $G$ , $H$ also satisfies $α$ .
It is a quotient-closed group property, i.e., whenever $G$ is a group satisfying $α$ and $H$ is a normal subgroup of $G$ , the quotient group $G / H$ also satisfies $α$ .
It is a direct product-closed group property, i.e., whenever $G_{1}, G_{2}, \dots$ are groups all of which satisfy $α$ , the external direct product $G_{1} \times G_{2} \times \dots \times G_{n}$ for any $n$ , and the infinite direct product $G_{1} \times G_{2} \times \dots$ also satisfy $α$ .

Relation with other metaproperties

Weaker metaproperties

Metaproperty	Meaning	Intermediate notions
pseudovarietal group property	subgroup-closed, quotient-closed, and closed under finite direct products	\|FULL LIST, MORE INFO
subgroup-closed group property	closed under taking subgroups	\|FULL LIST, MORE INFO
quotient-closed group property	closed under taking quotient groups	\|FULL LIST, MORE INFO
direct product-closed group property	closed under taking direct products	\|FULL LIST, MORE INFO

@@ Line 3: / Line 3: @@
 ==Definition==
-A [[group property]] <math>\alpha</math> is termed '''Varietal''' if it satisfies the following three conditions:
+A [[group property]] <math>\alpha</math> is termed '''varietal''' if it satisfies the following three conditions:
 # It is a [[subgroup-closed group property]], i.e., whenever <math>G</matH> is a group satisfying <math>\alpha</math> and <math>H</math> is a subgroup of <math>G</math>, <math>H</math> also satisfies <math>\alpha</math>.
@@ Line 22: / Line 22: @@
 | [[Stronger than::quotient-closed group property]] || closed under taking [[quotient group]]s || || || {{intermediate notions short|quotient-closed group property|varietal group property}}
 |-
-| [[Stronger than::finite direct product-closed group property]] || closed under taking finite [[direct product]]s || || || {{intermediate notions short|finite direct product-closed group property|varietal group property}}
+| [[Stronger than::direct product-closed group property]] || closed under taking [[direct product]]s || || || {{intermediate notions short|direct product-closed group property|varietal group property}}
 |}