FC-group: Difference between revisions

Latest revision as of 04:19, 30 January 2013

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

Definition

Equivalent definitions in tabular format

No.	Shorthand	A group is termed a FC-group if ...	A group $G$ is termed a FC-group if ...
1	conjugacy classes are finite	every conjugacy class in it has finite size.	for every $x \in G$ , the conjugacy class of $x$ in $G$ is finite.
2	element centralizers have finite index	the centralizer of any element is a subgroup of finite index.	for any $x \in G$ , the index $[G : C_{G} (x)]$ of the centralizer $C_{G} (x)$ is finite.
3	finite subset centralizers have finite index	the centralizer of any finite subset is a subgroup of finite index.	for any finite subset $S \subseteq G$ , the index $[G : C_{G} (S)]$ of the centralizer $C_{G} (S)$ is finite.
4	finitely generated subgroup centralizers have finite index	the centralizer of any subgroup generated by a finite subset is of finite index.	for any finitely generated subgroup $H$ of $G$ , the index $[G : C_{G} (H)]$ is finite.

Metaproperties

Metaproperty name	Satisfied?	Statement with symbols
subgroup-closed group property	Yes	Suppose $G$ is a FC-group and $H$ is a subgroup of $G$ . Then, $H$ is also a FC-group.
quotient-closed group property	Yes	Suppose $G$ is a FC-group and $H$ is a normal subgroup of $G$ . Then, the quotient group $G / H$ is a FC-group.
finite direct product-closed group property	Yes	Suppose $G_{1}$ and $G_{2}$ are FC-groups. Then, so is the external direct product $G_{1} \times G_{2}$ .

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 abelian group works as a counterexample.	\|FULL LIST, MORE INFO
abelian group	all conjugacy classes have size one	obvious	any finite non-abelian group works as a counterexample.	\|FULL LIST, MORE INFO
FZ-group	the center has finite index	FZ implies FC	FC not implies FZ	\|FULL LIST, MORE INFO
group with finite derived subgroup	the derived subgroup is finite	finite derived subgroup implies FC	FC not implies finite derived subgroup	\|FULL LIST, MORE INFO
BFC-group	there is a common bound on the sizes of all conjugacy classes		FC not implies BFC	\|FULL LIST, MORE INFO

Study of this notion

Mathematical subject classification

Under the Mathematical subject classification, the study of this notion comes under the class: 20F24

The subject classification 20F24 is used for FC-groups, and their generalizations.

External links

Definition links

@@ Line 1: / Line 1: @@
 {{group property}}
-{{variationof|finiteness (groups)}}
-{{finitarily tautological group property}}
-{{variationof|Abelianness}}
 ==Definition==
-===Symbol-free definition===
+===Equivalent definitions in tabular format===
-A [[group]] is said to be a '''FC-group''' if it satisfies the following equivalent conditions:
+{| class="sortable" border="1"
+! No. !! Shorthand !! A group is termed a FC-group if ... !! A group <math>G</math> is termed a FC-group if ...
+|-
+| 1 || conjugacy classes are finite || every [[conjugacy class]] in it has finite size. || for every <math>x \in G</math>, the conjugacy class of <math>x</math> in <math>G</math> is finite.
+|-
+| 2 || element centralizers have finite index || the [[centralizer]] of any element is a [[subgroup of finite index]]. || for any <math>x \in G</math>, the [[index of a subgroup|index]] <math>[G:C_G(x)]</math> of the [[centralizer]] <math>C_G(x)</math> is finite.
+|-
+| 3 || finite subset centralizers have finite index || the [[centralizer]] of any finite subset is a [[subgroup of finite index]]. || for any finite subset <math>S \subseteq G</math>, the [[index of a subgroup|index]] <math>[G:C_G(S)]</math> of the [[centralizer]] <math>C_G(S)</math> is finite.
+|-
+| 4 || finitely generated subgroup centralizers have finite index || the [[centralizer]] of any subgroup [[subgroup generated|generated by]] a finite subset is of finite index. || for any [[finitely generated group|finitely generated]] subgroup <math>H</math> of <math>G</math>, the [[index of a subgroup|index]] <math>[G:C_G(H)]</math> is finite.
+|}
-* Every [[conjugacy class]] in it is finite
+==Metaproperties==
-* The [[centralizer]] of any element is a [[subgroup of finite index]]
-===Definition with symbols===
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
-A [[group]] <math>G</math> is said to be an '''FC-group''' if for any element <math>x</math> in <math>G</math>, the following equivalent conditions are satisfied:
+|-
+| [[satisfies metaproperty::subgroup-closed group property]] || Yes || || Suppose <math>G</math> is a FC-group and <math>H</math> is a subgroup of <math>G</math>. Then, <math>H</math> is also a FC-group.
-* There are only finitely many elements in its [[conjugacy class]], that is, every element has only finitely many conjugates.
+|-
-* The [[centralizer]] <math>C_G(x)</math> has finite [[index of a subgroup|index]] in <math>G</math>, viz <math>[G:C_G(x)]</math> is finite.
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || || Suppose <math>G</math> is a FC-group and <math>H</math> is a [[normal subgroup]] of <math>G</math>. Then, the quotient group <math>G/H</math> is a FC-group.
+|-
+| [[satisfies metaproperty::finite direct product-closed group property]] || Yes || || Suppose <math>G_1</math> and <math>G_2</math> are FC-groups. Then, so is the [[external direct product]] <math>G_1 \times G_2</math>.
+|}
 ==Relation with other properties==
@@ Line 27: / Line 33: @@
 ===Stronger properties===
-* [[Finite group]]
+{| class="sortable" border="1"
-* [[Abelian group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[BFC-group]]
+|-
-* [[FZ-group]]
+| [[Weaker than::finite group]] || has only finitely many elements || obvious || any infinite abelian group works as a counterexample. || {{intermediate notions short|FC-group|finite group}}
+|-
-===Weaker properties===
+| [[Weaker than::abelian group]] || all conjugacy classes have size one || obvious || any finite non-abelian group works as a counterexample. || {{intermediate notions short|FC-group|abelian group}}
+|-
-==Metaproperties==
+| [[Weaker than::FZ-group]] || the center has finite index || [[FZ implies FC]] || [[FC not implies FZ]] || {{intermediate notions short|FC-group|FZ-group}}
+|-
-{{S-closed}}
+| [[Weaker than::group with finite derived subgroup]] || the [[derived subgroup]] is finite || [[finite derived subgroup implies FC]] || [[FC not implies finite derived subgroup]] || {{intermediate notions short|FC-group|group with finite derived subgroup}}
+|-
-Any subgroup of a FC-group is a FC-group. This follows from the fact that if <math>H \le G</math> are groups, and <math>C</math> is a conjugacy class in <math>H</math>, then all elements of <math>C</math> are conjugate in <math>G</math>, and hence <math>C</math> is contained inside a conjugacy class in <math>G</math>.
+| [[Weaker than::BFC-group]] || there is a common bound on the sizes of all conjugacy classes || || [[FC not implies BFC]] || {{intermediate notions short|FC-group|BFC-group}}
+|}
-{{DP-closed}}
-A direct product of FC-groups is an FC-group. This follows from the fact that the equivalence relation of being conjugate is closed under direct products.
 ==Study of this notion==