Group property: Difference between revisions

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This article is about a general term. A list of important particular cases (instances) is available at Category:Group properties

Definition

Symbol-free definition

A group property is a map from the collection of all groups to the two-element set (true, false) with the property that any two isomorphic groups get mapped to the same thing.

Caution

A group property must be decidable purely based on the abstract group structure, and should not be dependent on additional structure (like topology, or analytic or algebraic structure). Even if such additional structure occurs in the definition, it should have a universal or existential quantification associated to it.

Examples

Important examples of group properties

Being Abelian is a group property: a group is Abelian if any two elements in it commute. Every group either is Abelian or is not Abelian.

Other examples are being nilpotent, simple, finite. Being connected is not a group property because it requires additional structure on the group, and one could have isomorphic groups one of which is connected and the other is not.

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-#redirect [[Group property space]]
+{{wikilocal}}
+{{particularcases|[[:Category:Group properties]]}}
+==Definition==
+===Symbol-free definition===
+A '''group property''' is a map from the collection of all groups to the two-element set (true, false) with the property that any two isomorphic groups get mapped to the same thing.
+===Caution===
+A group property must be decidable purely based on the abstract group structure, and should ''not'' be dependent on additional structure (like topology, or analytic or algebraic structure). Even if such additional structure occurs in the definition, it should have a universal or existential quantification associated to it.
+==Examples==
+===Important examples of group properties===
+Being [[Abelian group|Abelian]] is a group property: a group is Abelian if any two elements in it commute. Every group either ''is Abelian'' or ''is not Abelian''.
+Other examples are being [[nilpotent group|nilpotent]], [[simple group|simple]], [[finite group|finite]]. Being ''connected'' is not a group property because it requires additional structure on the group, and one could have isomorphic groups one of which is connected and the other is not.