Theorem Quotient by a Closed Subgroup Exists
theorem admitted

Quotient by a Closed Subgroup Exists

Let \(G\) be a linear algebraic group over an algebraically closed field and let \(H\subseteq G\) be a closed algebraic subgroup. Then the homogeneous space \(G/H\) exists as a variety, the quotient map \[ \pi:G\to G/H \] is a morphism, and \(\pi\) is universal among morphisms from \(G\) that are constant on right \(H\)-cosets. The variety \(G/H\) is quasi-projective and \[ \dim(G/H)=\dim G-\dim H. \]

If \(H\) is normal, then \(G/H\) is a linear algebraic group and \(\pi\) is a homomorphism.

Proof

The construction embeds \(G\) into a finite-dimensional representation and chooses a point whose stabilizer is \(H\) after passing to a suitable projective representation. The orbit of that point is locally closed by the orbit theorem, and it represents the coset space \(G/H\).

The universal property follows because a morphism constant on \(H\)-cosets is constant on the fibers of \(G\to G/H\), hence factors uniquely through the orbit variety. The dimension formula is the fiber-dimension theorem applied to the quotient morphism, whose fibers are translates of \(H\). If \(H\) is normal, multiplication and inversion descend from \(G\) to \(G/H\), giving the quotient group structure.