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A transitive action on more than one point is primitive exactly when a point stabilizer is maximal

Statement

Let $G$ act transitively on $Ω$ with $∣ Ω ∣ > 1$ , and fix $α \in Ω$ . Then the action is primitive if and only if the point stabilizer $G_{α} : = {g \in G : g \cdot α = α}$ is a maximal proper subgroup of $G$ .

Facts & Assumptions

Given: A transitive action of $G$ on $Ω$ with $∣ Ω ∣ > 1$ and a point $α \in Ω$ .

[L1]

A transitive action is primitive exactly when every block is either a singleton or the whole set (Primitive and imprimitive transitive actions).

[L2]

Blocks containing $α$ correspond bijectively to subgroups $H$ with $G_{α} \leq H \leq G$ , by $H \mapsto H \cdot α$ and $B \mapsto G_{B}$ (Blocks containing a point correspond to intermediate subgroups).

Proof

technique · direct

1.1

Because $∣ Ω ∣ > 1$ , choose $β \in Ω$ with $β \neq α$ . Transitivity gives $g \in G$ with $g \cdot α = β$ , so $g \notin G_{α}$ . Hence $G_{α}$ is a proper subgroup of $G$ .

givenchoose

1.2

For the converse direction, suppose $G_{α}$ is maximal proper. Let $B$ be a block containing $α$ . By [L2], the corresponding subgroup $G_{B}$ satisfies $G_{α} \leq G_{B} \leq G$ , so maximality gives $G_{B} = G_{α}$ or $G_{B} = G$ . The first case gives $B = {α}$ and the second gives $B = Ω$ by [L2]. Hence every block containing $α$ is trivial.

2.1

Let $C$ be any block and choose $c \in C$ . By transitivity, choose $g \in G$ with $g \cdot c = α$ . Then $g \cdot C$ is a block containing $α$ , so step 1.2 makes it either ${α}$ or $Ω$ . Applying $g^{- 1}$ shows that $C$ is respectively a singleton or $Ω$ . Hence the action is primitive by [L1].

L1step 1.2choose

2.2

For the forward direction, suppose the action is primitive. By [L2], any subgroup $H$ with $G_{α} \leq H \leq G$ corresponds to a block containing $α$ . By [L1], that block is either ${α}$ or $Ω$ , so [L2] forces $H = G_{α}$ or $H = G$ . Together with step 1.1, this makes $G_{α}$ maximal proper.

L1L2step 1.1

3.1

Step 2.2 proves that primitivity implies maximality, while steps 1.2 and 2.1 prove the converse.

step 2.2step 1.2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)