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For surjective inverse systems in the pro-p setting, the Frattini subgroup commutes with the inverse limit

Statement

Let $G = {\underset{\leftarrow}{l i m}}_{i} G_{i}$ be a surjective inverse limit of finite $p$ -groups, with coordinate projections $π_{i} : G \to G_{i}$ . Then

$Φ (G) = {x \in G : π_{i} (x) \in Φ (G_{i}) for every i} ≅ {\underset{\leftarrow}{l i m}}_{i} Φ (G_{i}) .$

Facts & Assumptions

Given: A surjective inverse system of finite $p$ -groups with inverse limit $G$ .

[F1]

The Frattini subgroup is the intersection of maximal proper closed subgroups (The Frattini subgroup of a profinite group is the intersection of its maximal proper closed subgroups).

[L1]

In a profinite group, maximal proper closed subgroups are open (Every maximal proper closed subgroup of a profinite group is open).

[L2]

The inverse limit has coordinate projections $π_{i}$ (The inverse limit has its canonical coordinate projection maps).

Proof

technique · direct

1.1F1L1L2givenalgebra

By [F1] and [L1], $Φ (G)$ is the intersection of the maximal open subgroups of $G$ . If $M_{i}$ is a maximal subgroup of some finite quotient $G_{i}$ , then $π_{i}^{- 1} (M_{i})$ is a maximal open subgroup of $G$ . Conversely, every maximal open subgroup $U$ of $G$ contains the kernel of some coordinate projection, so $U = π_{i}^{- 1} (M_{i})$ for a maximal subgroup $M_{i} < G_{i}$ .

2.1F1step 1.1algebra∎

Therefore an element $x \in G$ lies in $Φ (G)$ exactly when $π_{i} (x)$ lies in every maximal subgroup of every finite quotient $G_{i}$ , that is, exactly when $π_{i} (x) \in Φ (G_{i})$ for every $i$ . The coordinatewise condition defines the inverse limit of the subgroups $Φ (G_{i})$ , so $Φ (G) ≅ {\underset{\leftarrow}{l i m}}_{i} Φ (G_{i})$ .

Depends on

Used by

For a finitely generated pro-p group, the Frattini subgroup is the closure of [G,G]Gᵖ Theorem

Dependency tree · two levels

5 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

Gareth Wilkes, Profinite Groups and Group Cohomology lecture notes (standard reference, not scraped)
Alexander Lubotzky, Combinatorial group theory for pro-p groups (standard reference, not scraped)