Minimal simple group

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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 p) every p-local subgroup is a p-solvable group
group in which every p-local subgroup is p-constrained every p-local subgroup is a p-constrained group

Facts