Double-coset sizes not dividing group order
= Double-coset sizes not dividing group order
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 $H=K=\langle(12)\rangle$. Then $HeH$ has size two, while $H(13)H$ has size four, which does not divide six. The general size formula $|HgK|=|H||K|/|H\cap gKg^{-1}|$ explains why different double cosets can have different sizes.