Acta Mathematica Sinica, English Series

, Volume 35, Issue 2, pp 204–212 | Cite as

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

  • He Guo LiuEmail author
  • Yu Lei Wang
  • Ji Ping Zhang


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 EE is a linear space over Q with dimension 2n and m + 2n = r. Furthermore, let AutG′G be the normal subgroup of AutG consisting of all elements of AutG which act trivially on the derived subgroup G′ of G, and AutGGGG 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 → AutG′G → AutG → AutG′ → 1 is split; (ii) AutG′G/AutGGGG ≅ Sp(2n,Q) × (GL(m, Q) ⋉ ℚ(m)); (iii) AutGGGG/InnG ≅ ℚ(2nm).


Extraspecial ℚ-group group extension symplectic group automorphism group 

MR(2010) Subject Classification

20E36 20E22 


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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHubei UniversityWuhanP. R. China
  2. 2.Department of MathematicsHe’nan University of TechnologyZhengzhouP. R. China
  3. 3.The School of Mathematical SciencesPeking UniversityBeijingP. R. China

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