Almost simple group: Difference between revisions

Latest revision as of 03:10, 17 December 2011

Definition

Symbol-free definition

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

There is a simple non-abelian group such that the given group can be embedded between the simple group and its automorphism group.
The group has a centralizer-free non-abelian simple normal subgroup.

Definition with symbols

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

There is a simple non-abelian group $S$ such that $S \leq T \leq A u t (S)$ for some group $T$ isomorphic to $G$ .
There exists a normal subgroup $N$ of $G$ such that $N$ is a simple non-abelian group and $C_{G} (N)$ is trivial.

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

This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

Relation with other properties

Stronger properties

Simple non-Abelian group

Facts

Automorphism group of simple non-Abelian group is complete
Almost simple not implies simple or complete: An almost simple group need not be either simple or complete: in other words, it can be properly sandwiched between a simple group and its automorphism group.
Symmetric groups are almost simple: For $n \geq 5$ , the symmetric group on $n$ letters is almost simple. Note that for $n \neq 6$ , it is in fact the whole automorphism group.

@@ Line 1: / Line 1: @@
-{{group property}}
+[[importance rank::3| ]]
-{{variationof|simplicity}}
 ==Definition==
 ===Symbol-free definition===
-A [[group]] is said to be '''almost simple''' if there is a [[simple group|simple]] non-Abelian group such that the given group can be embedded between the simple group and its automorphism group.
+A [[group]] is said to be '''almost simple''' if it satisfies the following equivalent conditions:
+* There is a [[simple non-abelian group]] such that the given group can be embedded between the simple group and its automorphism group.
+* The group has a [[centralizer-free subgroup|centralizer-free]] non-abelian [[simple normal subgroup]].
 ===Definition with symbols===
-A [[group]] <math>G</math> is said to be '''almost simple''' if there is a [[simple group|simple]] non-Abelian group <math>S</math> such that <math>S</math> &le; <math>G</math> &le; <math>Aut(S)</math>.
+A [[group]] <math>G</math> is said to be '''almost simple''' if it satisfies the following equivalent conditions:
+* There is a [[simple group|simple]] non-abelian group <math>S</math> such that <math>S \le T \le \operatorname{Aut}(S)</math> for some group <math>T</math> isomorphic to <math>G</math>.
+* There exists a normal subgroup <math>N</math> of <math>G</math> such that <math>N</math> is a simple non-abelian group and <math>C_G(N)</math> is trivial.
+{{group property}}
+{{variationof|simplicity}}
 ==Relation with other properties==
@@ Line 18: / Line 22: @@
 * [[Simple non-Abelian group]]
+==Facts==
+* [[Automorphism group of simple non-Abelian group is complete]]
+* [[Almost simple not implies simple or complete]]: An almost simple group need not be either simple or complete: in other words, it can be properly sandwiched between a simple group and its automorphism group.
+* [[Symmetric groups are almost simple]]: For <math>n \ge 5</math>, the symmetric group on <math>n</math> letters is almost simple. Note that for <math>n \ne 6</math>, it is in fact the whole automorphism group.