# Minimal simple group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

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.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Minimal simple group, all facts related to Minimal simple group) |Survey articles about this | Survey articles about definitions built on this

VIEW RELATED: Analogues of this | Variations of this | Opposites of this |

View a list of other standard non-basic definitions

## 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 |