In group theory, a double coset is a concept associated with a group acting on itself in a specific way. More formally, if \( G \) is a group and \( H \) and \( K \) are two subgroups of \( G \), the double coset of \( H \) and \( K \) with respect to an element \( g \in G \) is denoted by \( HgK \).
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For subgroups , the double coset of isThe double cosets partition : they are the group orbits of left multiplication by and right multiplication by . In the group action of on the left cosets of , the stabilizer subgroup of is . Thus contains left cosets of .
For finite , the orbit-stabilizer theorem gives the numerical formulaThe index statement remains meaningful for infinite groups, without dividing infinite cardinalities.