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

Automorphisms of Extensions of Q by a Direct Sum of Finitely Many Copies of Q
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摘要 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 Aut G = 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/ζ E is a linear space over Q with dimension 2 n and m + 2 n = r. Furthermore, let Aut_G'G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and Aut_(G/ζG),_(ζG)G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G')G→ Aut G→ Aut G'→ 1 is split;(ii)Aut_(G')G/Aut_(G/ζG),_(ζG)G = Sp(2 n, Q) ×(GL(m, Q) Q^(m));(iii) Aut_(G/ζG),ζGG/Inn G= Q^(2 nm). 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 Aut G = 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/ζ E is a linear space over Q with dimension 2 n and m + 2 n = r. Furthermore, let Aut_G'G be the normal subgroup of Aut G consisting of all elements of Aut G which act trivially on the derived subgroup G of G, and Aut_(G/ζG),_(ζG)G be the normal subgroup of Aut G consisting of all central automorphisms of G which also act trivially on the center ζ G of G. Then(i) The extension 1→ Aut_(G')G→ Aut G→ Aut G'→ 1 is split;(ii)Aut_(G')G/Aut_(G/ζG),_(ζG)G = Sp(2 n, Q) ×(GL(m, Q) Q^(m));(iii) Aut_(G/ζG),ζGG/Inn G= Q^(2 nm).
出处 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2019年第2期204-212,共9页 数学学报(英文版)
基金 Supported by NSFC(Grant Nos.11771129 and 11601121) He'nan Provincial Natural Science Foundation of China(Grant No.162300410066)
关键词 Extraspecial Q-group GROUP extension SYMPLECTIC GROUP AUTOMORPHISM GROUP Extraspecial Q-group group extension symplectic group automorphism group
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