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Abelian normal subgroups of faithful primitive actions are regular

Statement

Let $G$ act faithfully and primitively on $Ω$ , and let $N ⊴ G$ be a nontrivial abelian normal subgroup. Then the action of $N$ on $Ω$ is regular.

Facts & Assumptions

Given: A faithful primitive action of $G$ on $Ω$ and a nontrivial abelian normal subgroup $N ⊴ G$ .

[L1]

In a faithful primitive action, every nontrivial normal subgroup is transitive (Normal subgroups of a primitive action are transitive or lie in the kernel).

[L2]

An action is regular exactly when it is both transitive and free (Regular actions).

Proof

technique · direct

1.1

By [L1], the action of $N$ on $Ω$ is transitive.

2.1

Fix $α \in Ω$ , and suppose $n \in N$ fixes $α$ . For any $β \in Ω$ , step 1.1 gives $m \in N$ with $β = m \cdot α$ . Since $N$ is abelian, $n \cdot β = n \cdot (m \cdot α) = (n m) \cdot α = (m n) \cdot α = m \cdot (n \cdot α) = m \cdot α = β .$

step 1.1givenchoose

3.1

Step 2.1 shows that any element of $N$ fixing one point fixes every point. Faithfulness of the ambient action therefore forces that element to be the identity. So the action of $N$ is free.

step 2.1

4.1

Steps 1.1 and 3.1 make the action of $N$ transitive and free, hence regular by [L2].

step 1.1step 3.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

J. S. Milne, Group Theory, Chapter 4 (standard reference, not scraped)
K. Conrad, Transitive Group Actions (standard reference, not scraped)