N-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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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

Facts

• Finite N-group is solvable or almost simple
• Classification of finite N-groups

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)

References

• Nonsolvable finite groups all of whose local subgroups are solvable by John Griggs Thompson, Bulletin of the American Mathematical Society, ISSN 10889485 (electronic), ISSN 02730979 (print), Volume 74, Page 383 - 437(Year 1968): In this paper (appearing across multiple issues of the Pacific Journal of Mathematics), Thompson classified all N-groups.^{WeblinkMore info}