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The O'Nan-Scott classification of finite primitive groups
Statement
Every finite primitive permutation group of degree at least belongs to exactly one of the five coarse O'Nan-Scott types used on this page: affine, almost simple, diagonal, product action, or twisted wreath.
Facts & Assumptions
Given: A finite primitive permutation group of degree at least .
In a primitive action of degree at least , every nontrivial normal subgroup is transitive and the socle is a direct product of one or two minimal normal subgroups.
The finite O'Nan-Scott analysis organizes exactly those socle patterns into the five coarse families used on this page.
The local items on this page define those five types and explain their socle data (Affine, almost simple, diagonal, product action, and twisted wreath types).
Proof
Because the action has degree at least , it is nontrivial. The socle analysis [A1] therefore applies to and reduces the action to the structure of one or two minimal normal subgroups.
The source theorem [A2] says that those socle configurations fall into exactly five families, and [L1] records the names and defining data of those families in the convention used here. Hence belongs to exactly one of the five listed types.
Depends on
- A finite primitive group has at most two minimal normal subgroups
- Affine, almost simple, diagonal, product action, and twisted wreath types
- Almost simple finite groups
- Product-action wreath products are primitive under the standard hypotheses
- A unique abelian minimal normal subgroup gives affine type
- This page uses the coarse five-type O'Nan-Scott convention
Used by
- A simple diagonal action Example
- FALSE: the O'Nan-Scott theorem is the classification of finite simple groups False statement
- FALSE: the O'Nan-Scott theorem requires the classification of finite simple groups False statement
- Finite 2-transitive groups have affine or almost simple socle type Proposition
- CFSG enters later refinements of the O'Nan-Scott reduction Remark
- The O'Nan-Scott theorem reduces finite primitive-group questions to socle types Remark
Dependency tree · two levels
19 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Leonard H. Soicher, Primitive permutation groups, section 'The O'Nan-Scott theorem' (standard reference, not scraped)