CA-group: Difference between revisions

Latest revision as of 23:40, 7 July 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

DISJUNCTIVE DEFINITION: This article defines a term where the definition (or at least, one of the equivalent definitions) has the form "every A is a B or a C."

History

Origin

The concept and terminology of CA-groups was introduced by Michio Suzuki in his attempts to solve Burnside's conjecture (which later became the odd-order theorem).

Definition

Below are listed some equivalent definitions of CA-group, also called centralizer is abelian group:

No.	Shorthand	A group is termed a CN-group if ...	A group $G$ is termed a CN-group if ...
1	element centralizers are abelian	the centralizer of any non-identity element is an abelian subgroup.	for every non-identity element $x\in G$ , the centralizer $C_{G}(x)$ (i.e., the set of all elements of $G$ that commute with $x$ ) is an abelian subgroup of $G$ .
2	subgroup centralizers are abelian	the centralizer of any nontrivial subgroup is an abelian subgroup.	for every nontrivial subgroup $H$ of $G$ , the centralizer $C_{G}(H)$ is an abelian subgroup of $G$ .
3	subgroups: abelian or centerless	every subgroup of the group is either an abelian group or a centerless group.	for every nontrivial subgroup $H$ of $G$ , either $H$ is abelian or $H$ is centerless, i.e., the center of $H$ is trivial.

Equivalence of definitions

Further information: equivalence of definitions of CA-group

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	direct from version (3) of the definition	Suppose $G$ is a CA-group and $H$ is a subgroup of $G$ . Then, $H$ is also a CA-group.
direct product-closed group property	Yes		Suppose $G_{i},i\in I$ are CA-groups. Then, the external direct product $\prod _{i\in I}G_{i}$ is also a CA-group.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian group		(by definition)	CA not implies abelian	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
CN-group				\|FULL LIST, MORE INFO

@@ Line 1: / Line 1: @@
 {{group property}}
+{{disjunctive definition}}
-{{centralizer-determined|[[Abelian group]]}}
 ==History==
@@ Line 11: / Line 10: @@
 ==Definition==
-===Symbol-free definition===
+Below are listed some '''equivalent definitions''' of CA-group, also called '''centralizer is abelian''' group:
-A [[group]] is termed a '''CA-group''' or a '''Centralizer is Abelian''' group if it satisfies the following conditions:
+{| class="sortable" border="1"
+! No. !! Shorthand !! A group is termed a CN-group if ... !! A group <math>G</math> is termed a CN-group if ...
+|-
+| 1 || element centralizers are abelian || the [[defining ingredient::centralizer]] of any non-identity element is an [[defining ingredient::abelian group|abelian]] subgroup. || for every non-identity element <math>x \in G</math>, the [[centralizer]] <math>C_G(x)</math> (i.e., the set of all elements of <math>G</math> that commute with <math>x</math>) is an [[abelian group|abelian]] subgroup of <math>G</math>.
+|-
+| 2  || subgroup centralizers are abelian || the [[centralizer]] of any nontrivial subgroup is an [[abelian group|abelian]] subgroup. || for every nontrivial subgroup <math>H</math> of <math>G</math>, the centralizer <math>C_G(H)</math> is an [[abelian group|abelian]] subgroup of <math>G</math>.
+|-
+| 3 || subgroups: abelian or centerless || every [[subgroup]] of the group is either an [[abelian group]] or a [[defining ingredient::centerless group]]. || for every nontrivial subgroup <math>H</math> of <math>G</math>, either <math>H</math> is abelian or <math>H</math> is centerless, i.e., the [[defining ingredient::center]] of <math>H</math> is trivial.
+|}
-* The [[centralizer]] of any nontrivial element is an [[Abelian group|Abelian]] subgroup.
+===Equivalence of definitions===
-* The [[centralizer]] of any nontrivial subgroup is an [[Abelian group|Abelian]] subgroup.
-===Definition with symbols===
+{{further|[[equivalence of definitions of CA-group]]}}
-A [[group]] <math>G</math> is termed a CA-group or a '''Centralizer is Abelian''' group if it satisfies the following conditions:
+==Metaproperties==
-* For any <math>e \ne x \in G</math>, the group <math>C_G(x)</math> is [[Abelian group|Abelian]]
+{| class="sortable" border="1"
-* For any nontrivial subgroup <math>H \le G</math>, the group <math>C_G(H)</math> is [[Abelian group|Abelian]]
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::subgroup-closed group property]] || Yes || direct from version (3) of the definition || Suppose <math>G</math> is a CA-group and <math>H</math> is a subgroup of <math>G</math>. Then, <math>H</math> is also a CA-group.
+|-
+| [[satisfies metaproperty::direct product-closed group property]] || Yes || || Suppose <math>G_i, i \in I</math> are CA-groups. Then, the [[external direct product]] <math>\prod_{i \in I} G_i</math> is also a CA-group.
+|}
 ==Relation with other properties==
@@ Line 29: / Line 40: @@
 ===Stronger properties===
-* [[Abelian group]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::abelian group]] || || (by definition) || [[CA not implies abelian]] || {{intermediate notions short|CA-group|abelian group}}
+|}
 ===Weaker properties===
-* [[CN-group]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-==Metaproperties==
+|-
+| [[Stronger than::CN-group]] || || || || {{intermediate notions short|CN-group|CA-group}}
-{{S-closed}}
+|}
-If <math>G</math> is a CA-group and <math>H \le G</math> is a subgroup, then for any <math>x \in H</math>, <math>C_H(x) = H \cap C_G(x)</math>. Hence, if <math>C_G(x)</math> is Abelian, so is <math>C_H(x)</math>. Thus, any subgroup of a CA-group is a CA-group.
-{{DP-closed}}
-If <math>G_1</math> and <math>G_2</math> are CA-groups, so is <math>G_1 \times G_2</math>, because the centralizer of the element <math>(x_1,x_2) \in G_1 \times G_2</math> is <math>C_{G_1}(x_1) \times C_{G_2}(x_2)</math>.