# N-group

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 properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

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 RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions

The term N-group is also used for a group satisfying the normalizer condition. Note that this meaning is entirely different

## History

The notion of N-group was studied extensively, and all finite N-groups were classified, in a monumental paper by John G. Thompson, titled Nonsolvable finite groups all of whose local subgroups are solvable.

## Definition

A group is termed an N-group if it satisfies the following equivalent conditions:

No. Shorthand A group is termed a N-group if ... A group $G$ is termed a N-group if ...
1 local implies solvable every local subgroup (i.e., the normalizer of a nontrivial solvable subgroup) in it is solvable. for any nontrivial solvable subgroup $Q$ of $G$, the normalizer $N_G(Q)$ is also solvable.
2 every subgroup is solvable or Fitting-free every subgroup of the group is either a solvable group or a Fitting-free group. for every subgroup $H$ of $G$, $H$ is either a solvable group or a Fitting-free group (i.e., a group with no nontrivial solvable normal subgroup).

### Equivalence of definitions

Further information: equivalence of definitions of N-group

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
solvable group the whole group is solvable
minimal simple group the group is a simple non-abelian group and every proper subgroup is solvable N-group not implies solvable or minimal simple

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
group in which every p-local subgroup is p-solvable for any prime number $p$ every p-local subgroup is a p-solvable group (by definition)
group in which every p-local subgroup is p-constrained for any prime number $p$ every p-local subgroup is a p-constrained group. (via every p-local subgroup is p-solvable)