ON EXTENSIONS OF ALGEBRAIC GROUPS WITH FINITE QUOTIENT

Consider an exact sequence of group schemes of finite type over a field k, 1 -> N -> G -> Q -> 1, where Q is finite. We show that Q lifts to a finite subgroup scheme F of G; if Q is etale and k is perfect, then F may be chosen etale as well. As applications, we obtain generalizations of classical results of Arima, Chevalley, and Rosenlicht to possibly nonconnected algebraic groups. We also show that every homogeneous space under such a group has a projective equivariant compactification.