Centerless group: Difference between revisions

Latest revision as of 21:48, 20 January 2013

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

Symbol-free definition

A group is said to be centerless if it satisfies the following equivalent conditions:

The center of the group is trivial.
The natural homomorphism from the group to its automorphism group that sends each element to the conjugation it induces, is injective (viz no two elements induce the same inner automorphism)

Definition with symbols

A group $G$ is said to be centerless if it satisfies the following equivalent conditions:

The center $Z (G)$ is the trivial group.
The natural homomorphism $G \to A u t (G)$ given by $g \mapsto c_{g}$ , where $c_{g} = x \mapsto g x g^{- 1}$ , is an injective map from $G$ to $A u t (G)$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
simple non-abelian group	non-abelian and simple: has no proper nontrivial normal subgroup	simple and non-abelian implies centerless	symmetric group:S3 is a centerless group that is not simple.	\|FULL LIST, MORE INFO
almost simple group	in between a simple non-abelian group and its automorphism group	almost simple implies centerless	symmetric group:S3 is a centerless group that is not almost simple.	\|FULL LIST, MORE INFO
complete group	centerless and every automorphism is inner	(by definition)	alternating group:A4 is centerless but not complete, it has outer automorphisms realized as conjugations via its embedding as A4 in S4.	\|FULL LIST, MORE INFO
characteristically simple non-abelian group	non-abelian and characteristically simple: has no proper nontrivial characteristic subgroup	characteristically simple and non-abelian implies centerless	symmetric group:S3 is a centerless group that is not simple.	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Intermediate notions
capable group	can be expressed as (up to isomorphism) the inner automorphism group of some group; equivalently, the epicenter is trivial.	centerless implies capable -- basically, it is isomorphic to its own inner automorphism group	\|FULL LIST, MORE INFO
stem group	the center is contained in the derived subgroup.		\|FULL LIST, MORE INFO
unicentral group	epicenter equals center.		\|FULL LIST, MORE INFO

Metaproperties

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of centerless groups is centerless. This follows from the general fact that the center of a direct product equals the direct product of the individual centers.

@@ Line 10: / Line 10: @@
 * The center of the group is [[trivial group|trivial]].
-* The natural homomorphism from the group to its automorphism group that sends each element to the conjugation it induces, is injective (viz no two elements induce the same inner automorphism)
+* The [[group acts as automorphisms by conjugation|natural homomorphism from the group to its automorphism group]] that sends each element to the conjugation it induces, is injective (viz no two elements induce the same inner automorphism)
 ===Definition with symbols===
@@ Line 17: / Line 17: @@
 * The center <math>Z(G)</math> is the trivial group.
-* The natural homomorphism <math>G \to Aut(G)</math> given by <math>g \mapsto c_g</math>, where <math>c_g = x \mapsto gxg^{-1}</math>, is an ''injective'' map from <math>G</math> to <math>Aut(G)</math>.
+* The natural homomorphism <math>G \to \operatorname{Aut}(G)</math> given by <math>g \mapsto c_g</math>, where <math>c_g = x \mapsto gxg^{-1}</math>, is an ''injective'' map from <math>G</math> to <math>\operatorname{Aut}(G)</math>.
 ==Relation with other properties==
@@ Line 40: / Line 40: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::capable group]] || can be expressed as (up to isomorphism) the [[inner automorphism group]] of some group || [[centerless implies capable]] -- basically, it is isomorphic to its own inner automorphism group || || {{intermediate notions short|capable group|centerless group}}
+| [[Stronger than::capable group]] || can be expressed as (up to isomorphism) the [[inner automorphism group]] of some group; equivalently, the [[epicenter]] is trivial. || [[centerless implies capable]] -- basically, it is isomorphic to its own inner automorphism group || || {{intermediate notions short|capable group|centerless group}}
 |-
-| [[Stronger than::stem group]] || the [[centeri]] is contained in the [[derived subgroup]]. || || || {{intermediate notions short|stem group|centerless group}}
+| [[Stronger than::stem group]] || the [[center]] is contained in the [[derived subgroup]]. || || || {{intermediate notions short|stem group|centerless group}}
+|-
+| [[Stronger than::unicentral group]] || [[epicenter]] equals [[center]]. || || || {{intermediate notions short|unicentral group|centerless group}}
 |}