Finitely generated FZ-group: Difference between revisions

Latest revision as of 06:04, 30 January 2013

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and FZ-group
View other group property conjunctions OR view all group properties

Definition

Equivalent definitions in tabular format

No.	Shorthand	A group is termed a finitely generated FZ-group if ...	A group $G$ is termed a finitely generated FZ-group if ...
1	it is finitely generated and FZ	it is both a finitely generated group and a FZ-group.	$G$ has a finite generating set and its inner automorphism group is a finite group.
2	FZ and center is finitely generated abelian	it is a FZ-group and its center is a finitely generated abelian group.	$G / Z (G)$ is a finite group and $Z (G)$ is a finitely generated abelian group.
3	finitely generated and finite derived subgroup	it is a finitely generated group and also a group with finite derived subgroup, i.e., its derived subgroup is a finite group.	$G$ has a finite generating set and the derived subgroup $G^{'}$ is a finite group.
4	finitely generated and FC	it is a finitely generated group and a FC-group	$G$ has a finite generating set and every conjugacy class in $G$ is finite.

Equivalence of definitions

Implication direction	Proof
(1) implies (2)	Schreier's lemma gives that any subgroup of finite index in a finitely generated group is finitely generated, hence, the center is finitely generated.
(2) implies (1)	If the center is finitely generated, then so is the whole group, because the center has finite index, so we need to add in only finitely many elements to the generating set of the center to get a generating set of the whole group.
(1) implies (3)	follows from FZ implies finite derived subgroup (also known as the Schur-Baer theorem).
(3) implies (4)	follows from finite derived subgroup implies FC (straightforward).
(4) implies (1)	see finitely generated and FC implies FZ.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite group	has only finitely many elements	(obvious)	any infinite finitely generated abelian group provides a counterexample.	\|FULL LIST, MORE INFO
finitely generated abelian group	finitely generated and abelian	(obvious)	any finite non-abelian group provides a counterexample.	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
FZ-group	center has finite index	(obvious)	any abelian group that is not finitely generated gives a counterexample.	\|FULL LIST, MORE INFO
group with finite derived subgroup	derived subgroup is finite	(from definition)		\|FULL LIST, MORE INFO
BFC-group	there is a common finite bound on the size of all conjugacy classes			\|FULL LIST, MORE INFO
FC-group	every conjugacy class is finite			\|FULL LIST, MORE INFO

@@ Line 15: / Line 15: @@
 |-
 | 4 || finitely generated and FC || it is a [[finitely generated group]] and a [[defining ingredient::FC-group]] || <math>G</math> has a finite [[generating set of a group|generating set]] and every [[conjugacy class]] in <math>G</math> is finite.
+|}
+===Equivalence of definitions===
+{| class="sortable" border="1"
+! Implication direction !! Proof
+|-
+| (1) implies (2) || [[Schreier's lemma]] gives that any subgroup of finite index in a finitely generated group is finitely generated, hence, the center is finitely generated.
+|-
+| (2) implies (1) || If the center is finitely generated, then so is the whole group, because the center has finite index, so we need to add in only finitely many elements to the generating set of the center to get a generating set of the whole group.
+|-
+| (1) implies (3) || follows from [[FZ implies finite derived subgroup]] (also known as the Schur-Baer theorem).
+|-
+| (3) implies (4) || follows from [[finite derived subgroup implies FC]] (straightforward).
+|-
+| (4) implies (1) || see [[finitely generated and FC implies FZ]].
+|}
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::finite group]] || has only finitely many elements || (obvious) || any infinite [[finitely generated abelian group]] provides a counterexample. || {{intermediate notions short|finitely generated FZ-group|finite group}}
+|-
+| [[Weaker than::finitely generated abelian group]] || [[finitely generated group|finitely generated]] and [[abelian group|abelian]] || (obvious) || any finite non-abelian group provides a counterexample. || {{intermediate notions short|finitely generated FZ-group|finitely generated abelian group}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::FZ-group]] || center has finite index || (obvious) || any abelian group that is not finitely generated gives a counterexample. || {{intermediate notions short|FZ-group|finitely generated FZ-group}}
+|-
+| [[Stronger than::group with finite derived subgroup]] || [[derived subgroup]] is finite || (from definition) || || {{intermediate notions short|group with finite derived subgroup|finitely generated FZ-group}}
+|-
+| [[Stronger than::BFC-group]] || there is a common finite bound on the size of all [[conjugacy class]]es || || || {{intermediate notions short|BFC-group|finitely generated FZ-group}}
+|-
+| [[Stronger than::FC-group]] || every [[conjugacy class]] is finite || || || {{intermediate notions short|FC-group|finitely generated FZ-group}}
 |}