Double coset

For subgroups $H,P\le G$, the double coset of $x\in G$ isThe double cosets partition $G$. The action of $H$ on the left cosets of $P$ givesby the orbit-stabilizer theorem.

For the order-eight dihedral group generated by $r$ of order four and $s$ of order two with $srs=r^{-1}$, let $H=K=⟨s⟩$. There are exactly three double cosets: $H$, $H{r}^{2}H$, and $HrH$, of sizes two, two, and four. Their number does not divide eight. This differs from the number of ordinary cosets, which divides group order by Lagrange's theorem.

A double coset is generally not a subgroup, so Lagrange's theorem does not force its size to divide the ambient group order. In $S_3$, take $H=K=⟨(12)⟩$. Then $HeH$ has size two, while $H(13)H$ has size four, which does not divide six. The general size formula $|HgK|=|H||K|/|H\cap gK{g}^{-1}|$ explains why different double cosets can have different sizes.

For a group $G$ and subgroup $\Gamma$ such that every double coset has finitely many orbits under left multiplication by $\Gamma$, this algebra consists of complex $\Gamma$-bi-invariant functions on $G$ supported on finitely many double cosets. Its convolution isThe double coset indicator functions form a basis. On invariant modular functions it acts on the right by $f\ast h={\sum }_{\Gamma x}h(x)f{|}_{k}x$. For $G=G{L}_2(\mathbb{Q}{)}^{+}$ and $\Gamma =S{L}_2(\mathbb{Z})$, rational conjugation of finite-index modular subgroups verifies the finiteness condition. The determinant-$n$ Hecke operator uses the indicator of all integral determinant-$n$ matrices, not only a single double coset when $n$ is composite.

If $P$ is a Sylow subgroup of a finite group $G$ and $H\le G$, some intersection $H\cap xP{x}^{-1}$ is a Sylow subgroup of $H$. Otherwise every double-coset sizewould be divisible by a larger power of $p$ than $|G|$, as would their sum $|G|$.