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Normal subgroups of a primitive action are transitive or lie in the kernel

Statement

Let $G$ act primitively on $Ω$ , and let $N ⊴ G$ be a normal subgroup. Then either $N$ acts transitively on $Ω$ , or every element of $N$ fixes every point of $Ω$ . In other words, every normal subgroup of a primitive action is either transitive or contained in the action kernel.

In particular, if the action is faithful and $N \neq 1$ , then $N$ is transitive.

Facts & Assumptions

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

[L1]

A primitive action is a transitive action whose only block systems are the singleton partition and the one-block partition (Primitive and imprimitive transitive actions).

[L2]

Partitions into blocks are exactly the $G$ -invariant equivalence relations (G-invariant block systems are exactly the invariant equivalence relations).

[L3]

A normal subgroup satisfies $g N g^{- 1} = N$ for every $g \in G$ (Normal subgroup: invariance under conjugation).

Proof

technique · direct

1.1

Define $x \sim_{N} y$ when $y = n \cdot x$ for some $n \in N$ . This is an equivalence relation because $1 \in N$ , inverses in $N$ reverse the relation, and products in $N$ compose it.

givenalgebra

2.1

The relation $\sim_{N}$ is $G$ -invariant: if $y = n \cdot x$ with $n \in N$ , then for every $g \in G$ one has $g \cdot y = (g n g^{- 1}) \cdot (g \cdot x)$ , and [L3] puts $g n g^{- 1}$ back in $N$ .

L3step 1.1

3.1

By [L2], the $\sim_{N}$ -classes form a block system. Since the action is primitive, [L1] makes that block system either the one-block partition or the singleton partition.

L1L2step 2.1

4.1

In the one-block case, every point lies in the $N$ -orbit of every other point, so $N$ is transitive. In the singleton case, every $N$ -orbit has one point, so each $n \in N$ fixes every point of $Ω$ .

step 3.1

5.1

If the action is faithful and $N \neq 1$ , the second case of step 4.1 is impossible. Hence a nontrivial normal subgroup of a faithful primitive action is transitive.

step 4.1∎

Depends on

Used by

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)