DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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Double cosets $K \ G / H$ of two subgroups

Definition

Let $G$ be a group and let $K, H \leq G$ be subgroups (Subgroup). For $g \in G$ , the $(K, H)$ -double coset of $g$ is

K g H : = {k g h : k \in K, h \in H} .

The set of all such double cosets is written

K \ G / H .

Remarks

A double coset is a union of left cosets of $H$ and also a union of right cosets of $K$ .
The left cosets and right cosets of Left and right cosets $g H$ and $H g$ of a subgroup are the special cases $1 \ G / H$ and $K \ G / 1$ .

Depends on

Used by

Mackey's double-coset formula for restricting an induced character Theorem

Dependency tree · two levels

6 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

Peter Webb, A Course in Finite Group Representation Theory, Section 5.1 (standard reference, not scraped)
Anupam Singh, Representation Theory of Finite Groups, Section 20.1 (standard reference, not scraped)

Double cosets K\G/HK\G/H of two subgroups