Capable group: Difference between revisions

Latest revision as of 16:13, 19 November 2023

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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Definition

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

It is isomorphic to the inner automorphism group of some group. In other words, there is a group $H$ such that $G$ is isomorphic to the quotient group $H/Z(H)$ where $Z(H)$ is the center of the group.
Its epicenter is the trivial group.

In terms of the image operator

The group property of being a capable group is obtained by applying the image operator for the quotient-defining function sending each group to its inner automorphism group.

Facts

The trivial group is capable; it occurs as the inner automorphism group of any abelian group
A nontrivial cyclic group cannot be capable. This is because there cannot be an element outside the center of a group, which, along with the center, generates the whole group. For full proof, refer: cyclic and capable implies trivial
The quaternion group is not capable. For full proof, refer: quaternion group is not capable

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
centerless group	center is trivial	A centerless group is isomorphic to its own inner automorphism group.	The [Klein four-group]] is capable but not centerless.	\|FULL LIST, MORE INFO
simple non-abelian group	non-abelian and simple: no proper nontrivial normal subgroup	(via centerless)	(via centerless)	\|FULL LIST, MORE INFO
almost simple group	between a simple non-abelian group and its automorphism group	(via centerless)	(via centerless)	\|FULL LIST, MORE INFO
characteristically simple non-abelian group	non-abelian and characteristically simple: no proper nontrivial characteristic subgroup	(via centerless)	(via centerless)	\|FULL LIST, MORE INFO
complete group	centerless and every automorphism is inner	(via centerless)	(via centerless)	\|FULL LIST, MORE INFO

@@ Line 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+A [[group]] <math>G</math> is said to be '''capable''' if it satisfies the following equivalent conditions:
-A [[group]] is said to be '''capable''' if it occurs as the [[inner automorphism group]] of some group.
+# It is isomorphic to the [[defining ingredient::inner automorphism group]] of some group. In other words, there is a group <math>H</math> such that <math>G</math> is isomorphic to the [[quotient group]] <math>H/Z(H)</math> where <math>Z(H)</math> is the [[center]] of the group.
+# Its [[defining ingredient::epicenter]] is the [[trivial group]].
-===Definition with symbols===
+===In terms of the image operator===
+The group property of being a capable group is obtained by applying the [[image operator]] for the [[quotient-defining function]] sending each group to its inner automorphism group.
-A group <math>G</math> is said to be '''capable''' if there exists a group <math>H</math> such that <math>G = Inn(H) = H/Z(H)</math> where <math>Z(H)</math> is the [[center]] of <math>H</math>.
+==Facts==
-===In terms of the image operator===
+* The [[trivial group]] is capable; it occurs as the inner automorphism group of any [[abelian group]]
+* A nontrivial [[cyclic group]] cannot be capable. This is because there cannot be an element ''outside'' the center of a group, which, along with the center, generates the whole group. {{proofat|[[cyclic and capable implies trivial]]}}
+* The [[quaternion group]] is not capable. {{proofat|[[quaternion group is not capable]]}}
+==Relation with other properties==
+===Stronger properties===
-The group property of being a capable group is obtained by applying the [[image operator]] for the [[quotient-defining function]] sending each group to its inner automorphism group.
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::centerless group]] || center is trivial || A centerless group is isomorphic to its own inner automorphism group. || The [Klein four-group]] is capable but not centerless. || {{intermediate notions short|capable group|centerless group}}
+|-
+| [[Weaker than::simple non-abelian group]] || non-abelian and [[simple group|simple]]: no proper nontrivial [[normal subgroup]] || ([[simple and non-abelian implies centerless|via centerless]]) || (via centerless) || {{intermediate notions short|capable group|simple non-abelian group}}
+|-
+| [[Weaker than::almost simple group]] || between a simple non-abelian group and its automorphism group || ([[almost simple implies centerless|via centerless]]) || (via centerless) || {{intermediate notions short|capable group|almost simple group}}
+|-
+| [[Weaker than::characteristically simple non-abelian group]] || non-abelian and [[characteristically simple group|characteristically simple]]: no proper nontrivial [[characteristic subgroup]] || ([[characteristically simple and non-abelian implies centerless|via centerless]]) || (via centerless) || {{intermediate notions short|capable group|characteristically simple non-abelian group}}
+|-
+| [[Weaker than::complete group]] || centerless and [[group in which every automorphism is inner|every automorphism is inner]] || (via centerless) || (via centerless) || {{intermediate notions short|capable group|complete group}}
+|}