Perfect Groups

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Perfect Groups

Copyright (c) Derek Holt and W. Plesken, 1989
     ISBN 0-19-853559-7
     QA 171. H685 1989 512\'.22-dc 19
     LCC  88032444 CIP

PREFACE

CONTENTS

NOTATION

1  INTRODUCTION
     Exercises

2  PERFECT GROUPS WITH NONTRIVIAL FITTING SUBGROUP
     2.1 Elementary constructions
          2.1.1 The subdirect product of two groups
          2.1.2 Subdirect products with identified subgroups
          2.1.3 The subdirect product of a collection of groups
          2.1.4 Projective limits, profinite groups, and pro p-groups
          Exercises
     2.2 The graph of isomorphism types of finite groups
          2.2.1 Definitions and basic properties
               Exercises
          2.2.2 Interesting subgraphs
               Exercises
          2.2.3 Almost a primary decomposition
               Exercises
          2.2.4 Some primary components
               Exercises
          2.2.5 Some relevant representation theory
               Exercises
          2.2.6 Example of a full classification: elementary abelian 2-groups by A 5
               Exercises
     2.3 Using infinite perfect groups
          2.3.1 Maximal Frattini extensions
               Exercises
          2.3.2 Space groups
               Exercises
          2.3.3 Compact p-adic analytic groups
               Exercises
          2.3.4 Examples of p-adic groups giving extensions of p-groups by A5
               Exercises

3  SYSTEMATIC ENUMERATION OF FINITE PERFECT GROUPS
     3.1 Outline of the procedure
     3.2 The computation of irreducible modules
          Exercises
     3.3 Building up the class H # p
          3.3.1 Theoretical description
          3.3.2 Computational details

4  BASIC STRUCTURE AND ENUMERATION OF PERFECT SPACE GROUPS
     4.1 Structure of crystallographic and p-adic space groups
          4.1.1 Finite quotients
               Exercises
          4.1.2 Reducible and irreducible space groups
               Exercises
          4.1.3 Frattini extensions
               Exercises
     4.2 Algorithmic determination
          4.2.1 Representation as affine groups; finding the extensions
               Exercises
          4.2.2 Finding the lattices

5  TABLES OF FINITE PERFECT GROUPS
     5.1 Description of tables
          5.1.1 Global arrangement
          5.1.2 Symbols and names of groups
          5.1.3 Information about individual groups
     5.2 Index of tables
     5.3 Tables of finite perfect groups.
          1. Class A5#2. Perfect extensions of 2-groups by A5.
          2. Classes A5#3 and A521#3 .Perfect extensions of 3-groups by A5 and A5 2^1.
          3. Classes A5#5 and A521#5. Perfect extensions of 5-groups by A5 and A521
          4. Classes A5#7 and A521#7. Perfect extensions of 7-groups by A5 and A521.
          5. Other classes A5#p and A521#p. Perfect extensions of p-groups by A5 and A521, for p > 7.
          6. Classes A5#n and A521#n, where Inl > 1.
          7. Classes A524\'#3, A521x24\'#3, A524\'CN21#3, A524\'C21#3, andA 521x(24\'C21)#3. Perfect extensions of 3-groups by A524\',A521x24\', A524\'CN21, A524\'C21, and A521x(24\'C21).
          8. Class L3(2)#2. Perfect extensions of 2-groups by L3(2).
          9. Classes L3(2)#3 and L3(2)21#3.Perfect extensions of 3-groups by L3(2) and L3(2)21.
          10. Classes L3(2)#7 and L3(2)21#7.Perfect extensions of 7-groups by L3(2) and L3(2)21.
          11. Classes L3(2)#11 and L3(2)21#11.Perfect extensions of 11-groups by L3(2) and L3(2)21.
          12. Classes L3(2)#n and L3(2)21#n, where frtl> 1.
          13. Classes A6#2 and A6312.Perfect extensions of 2-groups by A6 and A631.
          14. Classes A6#3 and A621#3.Perfect extensions of 3-groups by A6 and A621.
          15. Classes A6#n and A621#7c, where un> 1.
          16. Class L2(8)#2. Perfect extensions of 2-groups by L2(8).
          17. Class L2(11)#2. Perfect extensions of 2-groups by L2(11).
          18. Classes L2(11)#3 and L2(11)21#3.Perfect extensions of 3-groups by L2(11) and L2(11)21.
          19. Classes L2(11)#11 and L2(11)21#11.Perfect extensions of 11-groups by L2(11) and L2(11)21.
          20. Classes L2(13)#13 and L2(13)21#13.Perfect extensions of 13-groups by L2(13) and L2(13)21.
          21. Class L2(17)#2. Perfect extensions of 2-groups by L2(17).
          22. Other groups L2(q) and SL(2,q) of order up to 106.
          23. Classes A7#2 and A731#2.Perfect extensions of 2-groups by A7 and A731.
          24. Class L3(3)#3. Perfect extensions of 3-groups by L3(3).
          25. Class U3(3)#2. Perfect extensions of 2-groups by U3(3).
          26. Class A8#2. Perfect extensions of 2-groups by A8 - L4(2).
          27. Classes L3(4)#2 and L3(4)31#2.Perfect extensions of 2-groups by L3(4) and L3(4)31.
          28. Other simple and quasisimple groups of order up to 106.
          29. Class (A5xA5)#2. Perfect extensions of 2-groups by A5xA5.
          30. Classes (A5xA5)#p, (A5xA5)21#p and (A521xA521)#p, where p = 3or 5. Perfect extensions of 3- and 5-groups by A5xA5, (A5xA5)2\'and A52\'xA521.
          31. Class (A5xL3(2))#2. Perfect extensions of 2-groups by A5xL3(2).
          32. Classes (A5xL3(2))#p, (A521xL3(2))#p and (A5xL3(2)21)#p, wherep = 3, 5 or 7. Perfect extensions of 3- 5- and 7-groups byA5xL3(2), A521xL3(2) and A5xL3(2)21.
          33. Classes (A5xA6)#2 and (A5xA6)31#2.Perfect extensions of 2-groups by A5xA6 and A5xA631.
          34. Class (L3(2)xL3(2))#2.Perfect extensions of 2-groups by L3(2)xL3(2).
          35. Class (A5xL2(8))#2. Perfect extensions of 2-groups by A5xL2(8).
          36. Class (A5xL2(11))#2. Perfect extensions of 2-groups by A5xL2(11)
          37. Classes (L3(2)xA6)#2 and (L3(2)xA631)#2.Perfect extensions of 2-groups by L3(2)xA6 and L3(2)xA63\'.
          38. Class (L3(2)xL2(8))#2.Perfect extensions of 2-groups by L3(2)xL2(8).
          39. Class (L3(2)xL2(11))#2.Perfect extensions of 2-groups by L3(2)xL2(11).
          40. Othdirect and central products of simple and quasisimple groups.
     5.4 The orders of perfect groups of order up to a million

6  TABLES OF PERFECT SPACE GROUPS
     6.1 Description of tables
          6.1.1 Basic terminology
          6.1.2 Contents of tables
          6.1.3 Description of a space group in the tables
          6.1.4 Guide to the tables
          6.1.5 Abbreviations for certain matrices in the tables
     6.2 Index of tables
     6.3 Tables of perfect space groups
          1. Perfect space groups with point group A5.
          2. Perfect space groups with point group A521 (= SL(2,5)).
          3. Perfect space groups with point group A524\'.
          4. Perfect space groups with point group A524\'CN21.
          5. Perfect space groups with point group A524E2\'.
          6. Perfect space groups with point group A5(24E21A)C21.
          7. Perfect space groups with point group A524\'A24\'.
          8. Perfect space groups with point group A5341.
          9. Perfect space groups with point group A5(24\'x34\').
          10. Perfect space groups with point group (A5NxA5N)21.
          11. Perfect space groups with point group L3(2).
          12. Perfect space groups with point group L3(2)21 (- SL(2,7)).
          13. Perfect space groups with point group L3(2)23 .
          14. Perfect space groups with point group L3(2)N23\'.
          15. Perfect space groups with point group L3(2)23\'E2\'.
          16. Perfect space groups with point group L3(2)23E23\'.
          17. Perfect space groups with point group L3(2)(23x23\')E21.
          18. Perfect space groups with point group L3(2)(23x23\')C2\'.
          19. Perfect space groups with point group L3(2)(23E23\'E)C21.
          20. Perfect space groups with point group L3(2)((23\'x231E)C23)C21.
          21. Perfect space groups with point group A6.
          22. Perfect space groups with point group A621 (= SL(2,9)).
          23. Perfect space groups with point group A631.
          24. Perfect space groups with point group A624E21.
          25. Perfect space groups with point group A6(24E21A)CN21.
          26. Perfect space groups with point group A6(24x24\')E21.
          27. Perfect space groups with point group A7.
          28. Perfect space groups with point group A721
          29. Perfect space groups with point group A726.
          30. Perfect space groups with point group A726CN21.
          31. Perfect space groups with point group A8.
          32. Perfect space groups with point group A821.
          33. Perfect space groups with point group A826E21.
          34. Perfect space groups with point group A826CN21.
          35. Perfect space groups with point group A9.
          36. Perfect space groups with point group A921
          37. Perfect space groups with point group A928.
          38. Perfect space groups with point group A10.
          39. Perfect space groups with point group A102821.
          40. Perfect space groups with point group A11
          41. Perfect space groups with point group L2(8).
          42. Perfect space groups with point group L2(8)28.
          43. Perfect space groups with point group L2(11).
          44. Perfect space groups with point group L2(11)21 (= SL(2,1 1)).
          45. Perfect space groups with point group L2(13).
          46. Perfect space groups with point group L2(13)21(= SL(2,13)).
          47. Perfect space groups with point group L2(17)21 (= SL(2,17)).
          48. Perfect space groups with point group M 11.
          49. Perfect space groups with point group U3(3).
          50. Perfect space groups with point group U4(2).
          51. Perfect space groups with point group U4(2)21 (= Sp(4,3)).
          52. Perfect space groups with point group Sp6(2).
          53. Perfect space groups with point group Sp6(2)21.
          54. Perfect space groups with point group O1(2)21.

7  MAPPING A FINITELY PRESENTED GROUP ONTO A GROUP IN THE TABLES
     7.1 Finding simple images of a finitely presented group
     7.2 Rewriting presentations
     7.3 Lifting epimorphisms
     7.4 An example
     7.5 Finding epimorphisms onto space groups

REFERENCES

APPENDIX BY W. HANRATH: CHARACTER TABLES OF SOME FACTOR GROUPS OF SPACE GROUPS
     Table of contents of microfiche

INDEX OF NOTATION

AUTHOR INDEX

SUBJECT INDEX