N-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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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 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 of , the normalizer 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 of , 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
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 | 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 | 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.^{Weblink}^{More info}