Socle: Difference between revisions

Latest revision as of 14:51, 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 can be defined as the subgroup generated by all minimal normal subgroups, i.e., the join of all minimal normal subgroups.

Formalisms

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

Socle of finite group is internal direct product of simple subnormal subgroups: Since minimal normal implies characteristically simple, the socle is generated by characteristically simple groups. In a finite group, a characteristically simple group is a direct product of simple groups, so the socle is generated by simple groups. It can further be shown that the socle is an internal direct product of a collection of simple subnormal subgroups.
Socle equals Omega-1 of center for nilpotent p-group: If $G$ is a group of prime power order, or more generally in a (possibly infinite) nilpotent p-group, the socle equals Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega_1(Z(G))} , i.e., the set of elements of prime order in the center of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} (along with the identity element). This follows from the fact that minimal normal implies central in nilpotent.
If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} is a finite solvable group, then the socle is a product of elementary abelian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} -groups for a collection of primes dividing the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} (though this may not include all primes dividing the order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} ). This follows from the fact that minimal normal implies elementary abelian in finite solvable. (When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Groups that are not solvable

Here, the socle is a product of simple groups, but we cannot say a priori whether it will comprise only simple abelian groups, simple non-abelian groups, or both.

Subgroup properties

Properties satisfied

Property	Meaning	Proof of satisfaction
normal subgroup	invariant under all inner automorphisms	socle is normal
characteristic subgroup	invariant under all automorphisms	socle is characteristic
strictly characteristic subgroup	invariant under all surjective endomorphisms	socle is strictly characteristic
normality-preserving endomorphism-invariant subgroup	invariant under all normality-preserving endomorphisms	socle is normality-preserving endomorphism-invariant
weakly normal-homomorph-containing subgroup	contains any image under a homomorphism from the subgroup to the whole group that sends normal subgroups of the whole group contained in the subgroup to normal subgroups of the whole group	socle is weakly normal-homomorph-containing
direct projection-invariant subgroup	invariant under all projections to direct factors	socle is direct projection-invariant
finite direct power-closed characteristic subgroup	in any finite direct power of the whole group, the corresponding power of the socle is characteristic	socle is finite direct power-closed characteristic

Properties not satisfied

These are properties that are not always satisfied by the socle. They may be satisfied by the socle in a particular group.

Property	Meaning	Proof of dissatisfaction
fully invariant subgroup	invariant under all endomorphisms	socle not is fully invariant
transitively normal subgroup	every normal subgroup of the subgroup is normal in the whole group	socle not is transitively normal

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 2: / Line 2: @@
 ==Definition==
-The '''socle''' of a group is defined as the subgroup generated by all [[minimal normal subgroup]]s.
+The '''socle''' of a group can be defined as the subgroup generated by all [[minimal normal subgroup]]s, i.e., the [[join of subgroups|join]] of all minimal normal subgroups.
+==Formalisms==
-==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.
 {{obtainedbyapplyingthe|join-all operator|minimal normal subgroup}}
@@ Line 13: / Line 10: @@
 ==Facts==
-* [[Socle equals Omega-1 of center in nilpotent p-group]]: 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]].
+* [[Socle of finite group is internal direct product of simple subnormal subgroups]]: Since [[minimal normal implies characteristically simple]], the socle is generated by [[characteristically simple group]]s. In a [[finite group]], a characteristically simple group is a direct product of simple groups, so the socle is generated by simple groups. It can further be shown that the socle is an internal direct product of a collection of simple subnormal subgroups.
+* [[Socle equals Omega-1 of center for nilpotent p-group]]: 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]]. (When <math>G</math> is nilpotent, ''all'' primes dividing its order are included).
@@ Line 30: / Line 28: @@
 {{#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}}
+===Groups that are not solvable===
+Here, the socle is a product of simple groups, but we cannot say a priori whether it will comprise only simple abelian groups, simple non-abelian groups, or both.
+{{#ask: [[arises as subgroup-defining function::socle]][[group part.dissatisfies property::solvable group]]|?group part|?subgroup part|?quotient part}}
+==Subgroup properties==
+===Properties satisfied===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of satisfaction
+|-
+| [[satisfies property::normal subgroup]] || invariant under all [[inner automorphism]]s || [[socle is normal]]
+|-
+| [[satisfies property::characteristic subgroup]] || invariant under all [[automorphism]]s || [[socle is characteristic]]
+|-
+| [[satisfies property::strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || [[socle is strictly characteristic]]
+|-
+| [[satisfies property::normality-preserving endomorphism-invariant subgroup]] || invariant under all [[normality-preserving endomorphism]]s || [[socle is normality-preserving endomorphism-invariant]]
+|-
+| [[satisfies property::weakly normal-homomorph-containing subgroup]] || contains any image under a homomorphism from the subgroup to the whole group that sends normal subgroups of the whole group contained in the subgroup to normal subgroups of the whole group || [[socle is weakly normal-homomorph-containing]]
+|-
+| [[satisfies property::direct projection-invariant subgroup]] || invariant under all projections to direct factors || [[socle is direct projection-invariant]]
+|-
+| [[satisfies property::finite direct power-closed characteristic subgroup]] || in any finite direct power of the whole group, the corresponding power of the socle is characteristic || [[socle is finite direct power-closed characteristic]]
+|}
+===Properties not satisfied===
+These are properties that are not ''always'' satisfied by the socle. They may be satisfied by the socle in a particular group.
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of dissatisfaction
+|-
+| [[dissatisfies property::fully invariant subgroup]] || invariant under all [[endomorphism]]s || [[socle not is fully invariant]]
+|-
+| [[dissatisfies property::transitively normal subgroup]] || every normal subgroup of the subgroup is normal in the whole group || [[socle not is transitively normal]]
+|}
 ==Computation==
 {{GAP command for sdf|Socle}}