This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
- 1 Definition
- 2 Formalisms
- 3 Facts
- 4 Examples
- 5 Subgroup properties
- 6 Computation
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
- 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 is a group of prime power order, or more generally in a (possibly infinite) nilpotent p-group, the socle equals , i.e., the set of elements of prime order in the center of (along with the identity element). This follows from the fact that minimal normal implies central in nilpotent.
- If is a finite solvable group, then the socle is a product of elementary abelian -groups for a collection of primes dividing the order of (though this may not include all primes dividing the order of ). This follows from the fact that minimal normal implies elementary abelian in finite solvable. (When is nilpotent, all primes dividing its order are included).
Groups of prime power order
Here, the socle is Omega-1 of the center:
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.
|Group part||Subgroup part||Quotient part|
|A5 in S5||Symmetric group:S5||Alternating group:A5||Cyclic group:Z2|
|Center of special linear group:SL(2,5)||Special linear group:SL(2,5)||Cyclic group:Z2||Alternating group:A5|
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|
The command for computing this subgroup-defining function in Groups, Algorithms and Programming (GAP) is:Socle