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On the Automorphisms of the Classical Groups.

By: Contributor(s): Material type: TextTextSeries: Memoirs of the American Mathematical SocietyPublisher: Providence : American Mathematical Society, 1951Copyright date: ©1951Edition: 1st edDescription: 1 online resource (130 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9780821899618
Subject(s): Genre/Form: Additional physical formats: Print version:: On the Automorphisms of the Classical GroupsLOC classification:
  • QA174.2.D54 1951
Online resources:
Contents:
Intro -- CONTENTS -- I. INTRODUCTION -- 1 . Survey of the paper -- 2. General lemmas -- II. AUTOMORPHISMS OF GL[sub(n)](K) (n ≥ 3, K sfield of characteristic ≠ 2) -- 3. Group-theoretic classification of involutions -- 4. Group-theoretic characterization of transvections -- 5. Determination of the automorphisms of GL[sub(n)](K) -- III. AUTOMORPHISMS OF PGL[sub(n)](K) (n ≥ 3, K sfield of characteristic ≠ 2) -- 6. Involutiona in PGL[sub(n)](K) -- 7. Group-theoretic characterization of involutiona of the firat kind -- 8. The case n=4 -- 9. The case n= 6 -- 10. Determination of the automorphisms of PGL[sub(n)](K) -- IV. AUTOMORPHISMS OF GL[sub(n)](K) AND PGL[sub(n)](K) (n ≥ 3, K sfield of characteristic 2) -- 11. Group-theoretic characterization of transvections -- 12. Determination of the automorphisms of GL[sub(n)](K) -- 13. Determination of the automorphisms of PGL[sub(n)](K) -- V. AUTOMORPHISMS OF SL[sub(n)](K) AND PSL[sub(n)](K) -- 14. Reduction to previous results -- 15. Group-theoretic characterization of transvections in SL[sub(n)](K) -- 16. Determination of the automorphisms of SL[sub(n)](K) -- 17. Determination of the automorphisms of PSL[sub(n)](K) -- 18. Isomorphisms between PSL[sub(n)](K) and PSL[sub(m)](K') -- VI. AUTOMORPHISMS OF Sp[sub(2m)](K) (K field of characteristic ≠ 2) -- 19. Group-theoretic characterization of the symplectic (n-2,2) and (2,n-2)-involutions -- 20. The case n=4 -- 21 . The case n=4 (continued) -- 22. Determination of the automorphisms of Sp[sub(n)](K) -- VII. AUTOMORPHISMS OF PSp[sub(n)](K) (K field of characteristic ≠ 2) -- 23. Group-theoretic characterization of involutions of the first kind -- 24. Determination of the automorphisms of PSp[sub(n)](K) -- VIII. AUTOMORPHISMS OF Sp[sub(2m)](K) (K field of characteristic 2) -- 25. Centralizers of involutions in Sp[sub(2m)](K).
26. Group-theoretic characterization of symplectic tranavections -- 27. The case n=4 -- 28. Determination of the automorphisms of Sp[sub(n)](K) -- 29. Isomorphisms between PSp[sub(2m)](K) and PSp[sub(2q)](K') -- IX. ISOMORPHISMS BETWEEN THE GROUPS PSp[sub(2m)](K) AND THE GROUPS PSL[sub(n)](K') AND V[sub(r)] -- 30. Isomorphisms between the groups PSp[sub(2m)](K) and PSL[sub(n)]K') -- 31 . Isomorphisms between the groups PSp[sub(2m)(K) and V[sub(n)] -- X. AUTOMORPHISMS OF O[sub(n)](K,f) (n ≥ 3, K field of characteristic ≠ 2, f quadratic from of index V ≥ 1) -- 32. Group-theoretic characterization of symmetriea -- 33. Determination of the automorphiama of O[sub(n)](K,f) -- 34. Determination of the automorphiama of O[sub(n)](K,f) (continued) -- XI. AUTOMORPHISMS OF O[sup(+)[sub(n)](K,f) (n ≥ 5, K field of characteristic ≠ 2, f quadratic from of index V ≥ 1) -- 35. First case : n odd -- 36. Second case : n even -- 37. Second case : n even (continued) -- XII. AUTOMORPHISMS OF PO[sub(n)](K,f) AND PO[sup(+)[sub(n)](K,f) (n even ≥ 4, K field of characteristic ≠ 2, f quadratic from of index V ≥ 1) -- 38. Characterization of the involutions of the fir at kind In PO[sub(n)](K,f) -- 39. Characterization of the involutiona of the firat kind In PO[sup(+)][sub(n)](K,f) -- 40. Characterization of the involutiona of the first kind In PO[sup(+)][sub(n)](K,f) (continued) -- XIII. AUTOMORPHISMS OF PΩ[sub(n)](K,.f ) (n ≥ 6, K finite field of characteristic ≠ 2) -- 41. Preliminaries -- 42. First case: n odd -- 43. Second case: n=2m even, f of square discriminant -- XIV. AUTOMORPHISMS OF PΩ[sub(n)](K,.f ) (n even ≥ 10, K finite field of characteristic 2) -- 44. Centralizers of involutions in PΩ[sub(n)](K,f) -- 45. Determination of the automorphiama of Ω[sub(n)](K,f) (n ≥ 10) -- 46. Determination of the automorphisms of Ω[sub(n)](K,f) (continued).
XV. ISOMORPHISMS BETWEEN THE GROUPS PΩ[sub(n)] (K,f') AND PSL[sub(n)](K'), PSp[sub(k)](K') OR V[sub(r)] (K finite field) -- 47. Isomorphisms between PΩ[sub(n)](K,f) and PΩ[sub(m)] (K',f') (K and K' of characteristic ≠ 2) -- 48. Isomorphisms between PΩ[sub(n)](K,f) and PSL[sub(m)(K'), or PSp[sub(m)(K'), or V[sub(r)] (K field of characteristic ≠ 2) -- 49. Isomorphisms between PΩ[sub(n)](K,f) and PΩ[sub(m)](K',f') (K field of characteristic 2) -- 50. Isomorphisms between PΩ[sub(n)](K,f) and PSL[sub(m)](K'), or PSp[sub(m)](K') of V[sub(r)] (K field of characteristic 2) -- XVI. AUTOMORPHISMS OF U[sub(n)](K,f) (n ≥ 3, K field of characteristic ≠ 2, f hermitian form of index V ≥ 1) -- 51. Characterization of symmetries in U[sub(n)](K,f) -- 52. Determination of the automorphisms of U[sub(n)](K,f) -- 53. Determination of the automorphisms of U[sub(n)](K,f) (continued) -- XVII. AUTOMORPHISMS OF U[sup(+)][sub(n)](K,f) (n ≥ 3, K afield of characteriatic ≠ 2, f hermitian form of index V ≥ 1) -- 54. Characterization of 2-involutions in U*[sub(n)](K,f) -- 55. Determination of the automorphisms of U[sup(+)][sub(n)](K,f) -- XVIII. AUTOMORPHISMS OF THE GROUP U[sub(n)](K,f) (n ≥ 3, K reflexive sfield of characteristic ≠ 2, f hermitian form of index V ≥ 1) -- 56. Characterization of symmetries In U[sub(n)](K,f) -- 57. Determination of the automorphisms of U[sub(n)](K,f) -- XIX. AUTOMORPHISMS OF PU[sup(+)][sub(n)](K) (n ≥ 3, K field of characteristic ≠ 2) -- 58. Characterization of involutions of the first kind -- XX. AUTOMORPHISMS OF PU[sup(+)][sub(n)](K) (n ≥ 3, K finite field of characteristic 2) -- 59. Centralizers of Involutions in PU[sup(+)][sub(n)](K) -- 60. Determination of the automorphisms of PU[sup(+)][sub(n)](K) -- 61 . Determination of the automorphisms of PU[sup(+)][sub(n)](K) (continued).
XXI. ISOMORPHISMS BETWEEN THE GROUPS PU[sup(+)][sub(n)](K) AND PSL[sub(n)](K'), P&amp -- #937[sub(m)](K',f') OR V[sub(r)] (K finite field) -- 62. Isomorphisms between PU[sup(+)][sub(n)](K) and PU[sup(+)][sub(m)](K') -- 63. Isomorphisms between PU[sup(+)][sub(n)](K) and PSL[sub(m)](K'), or PSp[sub(m)](K'), or PΩ[sub(m)](K',f'), or V[sub(r)] -- XXII. CONCLUSION -- 64. Unsolved problems -- SUPPLEMENT TO THE PAPER OF DIEUDONNÉ ON THE AUTOMORPHISMS OF CIASSICAL GROUPS -- 1 . Introduction -- I. LINEAR GROUPS -- 2. Automorphisms of the general linear group GL[sub(2)](K) -- 3. Automorphisms of SL[sup(±)][sub(2)](K) -- 4. Automorphisms of PGL[sub(2)](K) -- 5. Automorphisms of PSL[sup(±)][sub(2)](K) -- 6. Automorphism of SL[sub(4)](K) -- II. ORTHOGONAL GROUPS -- 7. Preliminaries -- 8. An extra type of automorphisms -- 9. Structure of the group O[sup(+)][sub(4)](K,f).
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Intro -- CONTENTS -- I. INTRODUCTION -- 1 . Survey of the paper -- 2. General lemmas -- II. AUTOMORPHISMS OF GL[sub(n)](K) (n ≥ 3, K sfield of characteristic ≠ 2) -- 3. Group-theoretic classification of involutions -- 4. Group-theoretic characterization of transvections -- 5. Determination of the automorphisms of GL[sub(n)](K) -- III. AUTOMORPHISMS OF PGL[sub(n)](K) (n ≥ 3, K sfield of characteristic ≠ 2) -- 6. Involutiona in PGL[sub(n)](K) -- 7. Group-theoretic characterization of involutiona of the firat kind -- 8. The case n=4 -- 9. The case n= 6 -- 10. Determination of the automorphisms of PGL[sub(n)](K) -- IV. AUTOMORPHISMS OF GL[sub(n)](K) AND PGL[sub(n)](K) (n ≥ 3, K sfield of characteristic 2) -- 11. Group-theoretic characterization of transvections -- 12. Determination of the automorphisms of GL[sub(n)](K) -- 13. Determination of the automorphisms of PGL[sub(n)](K) -- V. AUTOMORPHISMS OF SL[sub(n)](K) AND PSL[sub(n)](K) -- 14. Reduction to previous results -- 15. Group-theoretic characterization of transvections in SL[sub(n)](K) -- 16. Determination of the automorphisms of SL[sub(n)](K) -- 17. Determination of the automorphisms of PSL[sub(n)](K) -- 18. Isomorphisms between PSL[sub(n)](K) and PSL[sub(m)](K') -- VI. AUTOMORPHISMS OF Sp[sub(2m)](K) (K field of characteristic ≠ 2) -- 19. Group-theoretic characterization of the symplectic (n-2,2) and (2,n-2)-involutions -- 20. The case n=4 -- 21 . The case n=4 (continued) -- 22. Determination of the automorphisms of Sp[sub(n)](K) -- VII. AUTOMORPHISMS OF PSp[sub(n)](K) (K field of characteristic ≠ 2) -- 23. Group-theoretic characterization of involutions of the first kind -- 24. Determination of the automorphisms of PSp[sub(n)](K) -- VIII. AUTOMORPHISMS OF Sp[sub(2m)](K) (K field of characteristic 2) -- 25. Centralizers of involutions in Sp[sub(2m)](K).

26. Group-theoretic characterization of symplectic tranavections -- 27. The case n=4 -- 28. Determination of the automorphisms of Sp[sub(n)](K) -- 29. Isomorphisms between PSp[sub(2m)](K) and PSp[sub(2q)](K') -- IX. ISOMORPHISMS BETWEEN THE GROUPS PSp[sub(2m)](K) AND THE GROUPS PSL[sub(n)](K') AND V[sub(r)] -- 30. Isomorphisms between the groups PSp[sub(2m)](K) and PSL[sub(n)]K') -- 31 . Isomorphisms between the groups PSp[sub(2m)(K) and V[sub(n)] -- X. AUTOMORPHISMS OF O[sub(n)](K,f) (n ≥ 3, K field of characteristic ≠ 2, f quadratic from of index V ≥ 1) -- 32. Group-theoretic characterization of symmetriea -- 33. Determination of the automorphiama of O[sub(n)](K,f) -- 34. Determination of the automorphiama of O[sub(n)](K,f) (continued) -- XI. AUTOMORPHISMS OF O[sup(+)[sub(n)](K,f) (n ≥ 5, K field of characteristic ≠ 2, f quadratic from of index V ≥ 1) -- 35. First case : n odd -- 36. Second case : n even -- 37. Second case : n even (continued) -- XII. AUTOMORPHISMS OF PO[sub(n)](K,f) AND PO[sup(+)[sub(n)](K,f) (n even ≥ 4, K field of characteristic ≠ 2, f quadratic from of index V ≥ 1) -- 38. Characterization of the involutions of the fir at kind In PO[sub(n)](K,f) -- 39. Characterization of the involutiona of the firat kind In PO[sup(+)][sub(n)](K,f) -- 40. Characterization of the involutiona of the first kind In PO[sup(+)][sub(n)](K,f) (continued) -- XIII. AUTOMORPHISMS OF PΩ[sub(n)](K,.f ) (n ≥ 6, K finite field of characteristic ≠ 2) -- 41. Preliminaries -- 42. First case: n odd -- 43. Second case: n=2m even, f of square discriminant -- XIV. AUTOMORPHISMS OF PΩ[sub(n)](K,.f ) (n even ≥ 10, K finite field of characteristic 2) -- 44. Centralizers of involutions in PΩ[sub(n)](K,f) -- 45. Determination of the automorphiama of Ω[sub(n)](K,f) (n ≥ 10) -- 46. Determination of the automorphisms of Ω[sub(n)](K,f) (continued).

XV. ISOMORPHISMS BETWEEN THE GROUPS PΩ[sub(n)] (K,f') AND PSL[sub(n)](K'), PSp[sub(k)](K') OR V[sub(r)] (K finite field) -- 47. Isomorphisms between PΩ[sub(n)](K,f) and PΩ[sub(m)] (K',f') (K and K' of characteristic ≠ 2) -- 48. Isomorphisms between PΩ[sub(n)](K,f) and PSL[sub(m)(K'), or PSp[sub(m)(K'), or V[sub(r)] (K field of characteristic ≠ 2) -- 49. Isomorphisms between PΩ[sub(n)](K,f) and PΩ[sub(m)](K',f') (K field of characteristic 2) -- 50. Isomorphisms between PΩ[sub(n)](K,f) and PSL[sub(m)](K'), or PSp[sub(m)](K') of V[sub(r)] (K field of characteristic 2) -- XVI. AUTOMORPHISMS OF U[sub(n)](K,f) (n ≥ 3, K field of characteristic ≠ 2, f hermitian form of index V ≥ 1) -- 51. Characterization of symmetries in U[sub(n)](K,f) -- 52. Determination of the automorphisms of U[sub(n)](K,f) -- 53. Determination of the automorphisms of U[sub(n)](K,f) (continued) -- XVII. AUTOMORPHISMS OF U[sup(+)][sub(n)](K,f) (n ≥ 3, K afield of characteriatic ≠ 2, f hermitian form of index V ≥ 1) -- 54. Characterization of 2-involutions in U*[sub(n)](K,f) -- 55. Determination of the automorphisms of U[sup(+)][sub(n)](K,f) -- XVIII. AUTOMORPHISMS OF THE GROUP U[sub(n)](K,f) (n ≥ 3, K reflexive sfield of characteristic ≠ 2, f hermitian form of index V ≥ 1) -- 56. Characterization of symmetries In U[sub(n)](K,f) -- 57. Determination of the automorphisms of U[sub(n)](K,f) -- XIX. AUTOMORPHISMS OF PU[sup(+)][sub(n)](K) (n ≥ 3, K field of characteristic ≠ 2) -- 58. Characterization of involutions of the first kind -- XX. AUTOMORPHISMS OF PU[sup(+)][sub(n)](K) (n ≥ 3, K finite field of characteristic 2) -- 59. Centralizers of Involutions in PU[sup(+)][sub(n)](K) -- 60. Determination of the automorphisms of PU[sup(+)][sub(n)](K) -- 61 . Determination of the automorphisms of PU[sup(+)][sub(n)](K) (continued).

XXI. ISOMORPHISMS BETWEEN THE GROUPS PU[sup(+)][sub(n)](K) AND PSL[sub(n)](K'), P&amp -- #937[sub(m)](K',f') OR V[sub(r)] (K finite field) -- 62. Isomorphisms between PU[sup(+)][sub(n)](K) and PU[sup(+)][sub(m)](K') -- 63. Isomorphisms between PU[sup(+)][sub(n)](K) and PSL[sub(m)](K'), or PSp[sub(m)](K'), or PΩ[sub(m)](K',f'), or V[sub(r)] -- XXII. CONCLUSION -- 64. Unsolved problems -- SUPPLEMENT TO THE PAPER OF DIEUDONNÉ ON THE AUTOMORPHISMS OF CIASSICAL GROUPS -- 1 . Introduction -- I. LINEAR GROUPS -- 2. Automorphisms of the general linear group GL[sub(2)](K) -- 3. Automorphisms of SL[sup(±)][sub(2)](K) -- 4. Automorphisms of PGL[sub(2)](K) -- 5. Automorphisms of PSL[sup(±)][sub(2)](K) -- 6. Automorphism of SL[sub(4)](K) -- II. ORTHOGONAL GROUPS -- 7. Preliminaries -- 8. An extra type of automorphisms -- 9. Structure of the group O[sup(+)][sub(4)](K,f).

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