Automorphisms of Extensions of ℚ by a Direct Sum of Finitely Many Copies of ℚ

Abstract

Let G be an extension of ℚ by a direct sum of r copies of ℚ. (1) If G is abelian, then G is a direct sum of r + 1 copies of ℚ and AutG ≅ GL(r + 1, Q); (2) If G is non-abelian, then G is a direct product of an extraspecial ℚ-group E and m copies of ℚ, where E/ζE is a linear space over Q with dimension 2n and m + 2n = r. Furthermore, let Aut_G′G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G′ of G, and Aut_G/ζG,ζGG be the normal subgroup of AutG consisting of all central automorphisms of G which also act trivially on the center ζG of G. Then (i) The extension 1 → Aut_G′G → AutG → AutG′ → 1 is split; (ii) Aut_G′G/Aut_G/ζG,ζGG ≅ Sp(2n,Q) × (GL(m, Q) ⋉ ℚ^(m)); (iii) Aut_G/ζG,ζGG/InnG ≅ ℚ^(2nm).