Minimal simple group
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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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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Definition
Symbol-free definition
A group is said to be a minimal simple group if it is a simple non-abelian group and every proper subgroup of it is solvable.
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| simple non-abelian group | ||||
| N-group | every local subgroup is a solvable group | |||
| group in which every p-local subgroup is p-solvable (in the finite case, for any prime number ) | every -local subgroup is a p-solvable group | |||
| group in which every p-local subgroup is p-constrained | every -local subgroup is a p-constrained group |
