Socle: Difference between revisions

Revision as of 13:09, 8 July 2011

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

Definition

The socle of a group is defined as the subgroup generated by all minimal normal subgroups.

Group properties satisfied

The socle of a group is a direct product of simple groups. Further, any group that is the direct product of simple groups is its own socle.

In terms of the join-all operator

This property is obtained by applying the join-all operator to the property: minimal normal subgroup
View other properties obtained by applying the join-all operator

Facts

If $G$ is a group of prime power order, or more generally in a (possibly infinite) nilpotent p-group, the socle equals $\Omega _{1}(Z(G))$ , i.e., the set of elements of prime order in the center of $G$ (along with the identity element). This follows from the fact that minimal normal implies central in nilpotent.
If $G$ is a finite solvable group, then the socle is a product of elementary abelian $p$ -groups for a collection of primes dividing the order of $G$ (though this may not include all primes dividing the order of $G$ ). This follows from the fact that minimal normal implies elementary abelian in finite solvable. (When $G$ is nilpotent, all primes dividing its order are included).

Examples

Groups of prime power order

Here, the socle is Omega-1 of the center:

	Group part	Subgroup part	Quotient part
Center of dihedral group:D8	Dihedral group:D8	Cyclic group:Z2	Klein four-group

Finite solvable groups that are not nilpotent

Here, the socle is a product of elementary abelian groups for some of the primes dividing the order of the group:

Computation

GAP command

The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:Socle
View other GAP-computable subgroup-defining functions

@@ Line 14: / Line 14: @@
 * If <math>G</math> is a [[group of prime power order]], or more generally in a (possibly infinite) [[nilpotent p-group]], the socle equals <math>\Omega_1(Z(G))</math>, i.e., the set of elements of prime order in the [[center]] of <math>G</math> (along with the identity element). This follows from the fact that [[minimal normal implies central in nilpotent]].
-* If <math>G</math> is a [[finite solvable group]], then the socle is a product of elementary abelian <math>p</math>-groups for a collection of primes dividing the order of <math>G</math> (though this may not include ''all'' primes dividing the order of <math>G</math>). This follows from the fact that [[minimal normal implies elementary abelian in finite solvable]].
+* If <math>G</math> is a [[finite solvable group]], then the socle is a product of elementary abelian <math>p</math>-groups for a collection of primes dividing the order of <math>G</math> (though this may not include ''all'' primes dividing the order of <math>G</math>). This follows from the fact that [[minimal normal implies elementary abelian in finite solvable]]. (When <math>G</math> is nilpotent, ''all'' primes dividing its order are included).
 ==Examples==
@@ Line 24: / Line 24: @@
 {{#ask: [[arises as subgroup-defining function::socle]][[group part.satisfies property::group of prime power order]]|?group part|?subgroup part|?quotient part}}
+===Finite solvable groups that are not nilpotent===
+Here, the socle is a product of elementary abelian groups for some of the primes dividing the order of the group:
+{{#ask: [[arises as subgroup-defining function::socle]][[group part.satisfies property::solvable group]][[group part.dissatisfies property::nilpotent group]]|?group part|?subgroup part|?quotient part}}
 ==Computation==
 {{GAP command for sdf|Socle}}