Virtually abelian group: Difference between revisions

Latest revision as of 09:04, 14 February 2012

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of Abelianness|Find other variations of Abelianness |

Definition

Symbol-free definition

A group is said to be virtually Abelian if it has an Abelian subgroup of finite index.

Formalisms

In terms of the virtually operator

This property is obtained by applying the virtually operator to the property: Abelian group
View other properties obtained by applying the virtually operator

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication
finite group
abelian group
FZ-group	center is a subgroup of finite index	FZ implies virtually abelian
locally finite linear group over characteristic zero
metacyclic group

@@ Line 16: / Line 16: @@
 ===Stronger properties===
+{| class="sortable" border="1"
-* [[Finite group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Abelian group]]
+|-
-* [[Locally finite group|locally finite]] [[linear group]] over characteristic zero
+| [[Weaker than::finite group]] || || || ||
+|-
+| [[Weaker than::abelian group]] || || || ||
+|-
+| [[Weaker than::FZ-group]]|| [[center]] is a [[subgroup of finite index]]|| [[FZ implies virtually abelian]] || ||
+|-
+| [[Locally finite group|locally finite]] [[linear group]] over characteristic zero || || || ||
+|-
+| [[Weaker than::metacyclic group]] || || || ||
+|}