LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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The kernels of the finite coordinate projections form an open normal neighbourhood basis at the identity

Statement

For a profinite group presented as an inverse limit of finite discrete groups, the kernels of the coordinate projections form an open normal neighbourhood basis at the identity.

Facts & Assumptions

Given: A profinite group written as $L = \underset{\leftarrow}{l i m} G_{i}$ with coordinate maps $π_{i} : L \to G_{i}$ .

[L1]

Each $π_{i}$ is a coordinate projection from the inverse limit (The inverse limit has its canonical coordinate projection maps).

[F1]

Open sets in $L$ come from cylinders in the product topology, and a profinite group is such an inverse limit (The inverse limit of finite groups carries the subspace topology from the product of discrete factors, A profinite group is a topological group isomorphic to an inverse limit of finite discrete groups, Normal subgroup: invariance under conjugation, The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ).

Proof

technique · direct

1.1L1F1given

Each kernel $\ker π_{i}$ is a normal subgroup because $π_{i}$ is a homomorphism, and it is open because ${e_{i}} \subseteq G_{i}$ is open in the discrete factor and $π_{i}^{- 1} ({e_{i}}) = \ker π_{i}$ .

1.2F1L1givenchoose

Let $U$ be any neighbourhood of the identity in $L$ . By [F1], some basic cylinder fixing finitely many coordinates, say those in a finite set $F \subseteq I$ , is contained in $U$ . Choose $k \in I$ dominating every index in $F$ by directedness. If $x \in \ker π_{k}$ , then compatibility gives $π_{i} (x) = φ_{i k} (e_{k}) = e_{i}$ for every $i \in F$ , so $x$ lies in that cylinder. Hence $\ker π_{k} \subseteq U$ .

2.1step 1.1step 1.2∎

Therefore the family $(\ker π_{i})$ is a neighbourhood basis of open normal subgroups at the identity.

Depends on

Used by

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Sources

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)