Double coset-ordering subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed double coset-ordering if given any two double cosets $HxH$ and $HyH$ one cannot find $g,h,g',h'$ such that $gHxHg'\subset HyH$ and $hHyHh'\subset HxH$. In other words, the relation:

$HxH\leq HyH\iff \exists g,g'\qquad gHxHg'\subset HyH$

is a partial order.

Relation with other properties

Stronger properties

• Subgroup of double coset index two which is not normal