Algebraic Overline{ MathbbQ}-Groups As Abstract Groups. - Écon, Olivier (Memoirs of the American Mathematical Society Ser., 2018)
The author analyzes the abstract structure of algebraic groups over an algebraically closed field K. For K of characteristic zero and G a given connected affine algebraic \overline{\mathbb Q}-group, the main theorem describes all the affine algebraic \overline{\mathbb Q} -groups H such that the groups H(K) and G(K) are isomorphic as abstract groups. In the same time, it is shown that for any two connected algebraic \overline{\mathbb Q} -groups G and H, the elementary equivalence of the pure groups G(K) and H(K) implies that they are abstractly isomorphic. In the final section, the author appli.