Locally profinite group¶
In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup.
Core Idea¶
Locally profinite group is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup. Equivalently, a locally profinite group is a topological group that is Hausdorff, locally compact, and totally disconnected.
Scope of Application¶
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Hecke algebra of a locally profinite group. Let C^\inftyc(G) denote the space of locally constant functions on G with compact support.
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Examples. More generally, the matrix ring \operatorname{M}n(F) and the general linear group \operatorname{GL}n(F) are locally profinite.
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Examples. Another example of a locally profinite group is the absolute Weil group of a non-archimedean local field: this is in contrast to the fact that the absolute Galois group of such.
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Representations of a locally profinite group. Then a group homomorphism \psi: G \to \mathbb{C}^\times is continuous if and only if it has open kernel.
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Representations of a locally profinite group. We now make a blanket assumption that G/K is at most countable for all open compact subgroups K.
Clarity¶
A clear use of Locally profinite group names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup.
Manages Complexity¶
Locally profinite group compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—thus, we set \widetilde{V} = \bigcupK (V)K where K is acting through \rho^ and set \widetilde{\rho} = \rho^ .—and the practical consequence—then a group homomorphism \psi: G \to \mathbb{C}^\times is continuous if and only if it has open kernel.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In mathematics, a locally profinite group is a Hausdorff topological group in which every neighborhood of the identity element contains a compact open subgroup.
- Check operation and conditions. It is called the Hecke algebra of G and is denoted by \mathfrak{H}(G).
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Locally profinite group transfers literally when a new case preserves the same carrier type, relation, and recognition test. Let C^\inftyc(G) denote the space of locally constant functions on G with compact support. More generally, the matrix ring \operatorname{M}n(F) and the general linear group \operatorname{GL}n(F) are locally profinite. Beyond the home domain. No canonical parent is asserted for Locally profinite group.
Relationships to Other Abstractions¶
Current abstraction Locally profinite group Domain-specific
Parents (1) — more general patterns this builds on
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Locally profinite group is a kind of Group Prime
A locally profinite group is a group with a Hausdorff topology and a basis of compact open subgroups.
Hierarchy paths (5) — routes to 5 parentless roots
- Locally profinite group → Group → Monoid → Semigroup → Set and Membership
- Locally profinite group → Group → Monoid → Identity Element
- Locally profinite group → Group → Monoid → Semigroup → Closure
- Locally profinite group → Group → Monoid → Semigroup → Associativity → Invariance
- Locally profinite group → Group → Monoid → Semigroup → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Locally profinite group sits in a crowded region of the domain-specific corpus (35th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Algebraic Structures, Groups & Operators (42 abstractions)
Nearest neighbors
- Group Ring — 0.90
- Classifying space for SO(n) — 0.89
- Quasi-Frobenius Lie algebra — 0.88
- Change of Rings — 0.88
- Affiliated operator — 0.87
Computed from structural-signature embeddings · 2026-10-08