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A finite primitive group has at most two minimal normal subgroups

Statement

Let $G \leq Sym (Ω)$ be finite and primitive. Then $G$ has at most two minimal normal subgroups.

Facts & Assumptions

Given: A finite primitive permutation group $G \leq Sym (Ω)$ .

[L1]

Any two distinct minimal normal subgroups of $G$ are regular (Two distinct minimal normal subgroups of a primitive group are regular).

[L2]

Distinct minimal normal subgroups centralize one another (Distinct minimal normal subgroups centralize one another).

[A1]

A regular permutation group has exactly one element sending a chosen point to a chosen point.

Proof

technique · direct

1.1

Suppose that $M_{1}, M_{2}, M_{3}$ are three distinct minimal normal subgroups of $G$ . By [L1], each pair among them is regular. In particular, $M_{1}$ and $M_{2}$ are both regular.

givenL1

2.1

Fix $α \in Ω$ . Because $M_{1}$ is regular, the map $m \mapsto m α$ identifies $M_{1}$ with the set $Ω$ , and because $M_{2}$ is regular there is for each $β \in Ω$ a unique element $n (β) \in M_{2}$ with $n (β) α = β$ by [A1].

A1step 1.1choose

3.1

Applying [L2] to the pair $(M_{1}, M_{3})$ shows that $M_{3}$ centralizes $M_{1}$ . Hence every element of $M_{3}$ acts on the identified copy of $M_{1}$ by right translation. But $M_{2}$ already has that property by step 2.1, and the right-regular subgroup centralizing the left-regular action of $M_{1}$ is unique. Therefore $M_{3} = M_{2}$ , contradicting distinctness. So no third minimal normal subgroup exists.

L2step 2.1algebra∎

Depends on

Used by

A primitive group with two regular minimal normal subgroups Example
FALSE: every primitive group has a unique minimal normal subgroup False statement
A unique abelian minimal normal subgroup gives affine type Proposition
The O'Nan-Scott classification of finite primitive groups Theorem

Dependency tree · two levels

8 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 (standard reference, not scraped)