Groups of Prime Power Order. Volume 4
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Groups of Prime Power Order. Volume 4

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eBook - ePub

Groups of Prime Power Order. Volume 4

About this book

This is the fourth volume of a comprehensive and elementary treatment of finite p-group theory. As in the previous volumes, minimal nonabelian p-groups play an important role. Topics covered in this volume include:

  • subgroup structure of metacyclic p-groups
  • Ishikawa's theorem on p-groups with two sizes of conjugate classes
  • p-central p-groups
  • theorem of Kegel on nilpotence of H p-groups
  • partitions of p-groups
  • characterizations of Dedekindian groups
  • norm of p-groups
  • p-groups with 2-uniserial subgroups of small order

The book also contains hundreds of original exercises and solutions and a comprehensive list of more than 500 open problems. This work is suitable for researchers and graduate students with a modest background in algebra.

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Information

Publisher
De Gruyter
Year
2015
Print ISBN
9783110281453
eBook ISBN
9783110381559
Topic
Mathematics
Subtopic
Algebra
Index
Mathematics

§ 159 2-groups all of whose cyclic subgroups A, B with A η B = {1} generate an abelian subgroup

This and the following section were written by the second author. These sections contain the solution to Problem 2916.
Heng Lv, Wei Zhou and Dapeng Yu (see [LZY]) have been studying p-groups G all of whose cyclic subgroups are of index at most p2 in their normal closure in G. The main obstacle in these investigations has been two-generator subgroups {x, y} (x, y e G) such that 〈x〉 η 〈y〉 = {1} (see Lemma 144.4 and Lemma 144.5).
On the other hand, if G is a noncyclic p-group such that any two distinct maximal cyclic subgroups of G have intersection equal {1}, then we have either exp(G)= p or G = D2n, n > 3 (see Exercise 12 in Appendix 45). In fact, it is easy to prove this statement. Assume that exp(G) > p and let L G be a maximal cyclic subgroup of order > p. Let L M G with |M : L|= p. By Theorem 1.2, we get p = 2 and M is dihedral. Thus, all subgroups of G containing L and having order 2|L| are dihedral. Suppose that M G. Then N = NG(M) > M and N normalizes L. Let L0 be the cyclic subgroup of order 4 in L so that C = CN(L0) covers N/M, which gives C > L.Let K/L be a subgroup of order 2 in C/L. But then K is not dihedral, a contradiction. Hence we have M = G = D2n, n > 3, and we are done.
Therefore, it is of interest to classify p-groups G such that the following condition is satisfied:
image
In the first two Propositions 159.1 and 159.2 are determined p-groups G satisfying the condition (*), where G is either minimal nonabelian or metacyclic and p is any prime. It is clear that the condition (* ) says nothing in the case of p-groups of exponent p. But also in the case of 2-groups of exponent 4, the condition (*) says very little, and we characterize such groups in Theorem 159.3.
We recall that a p-group G is p-central if either p > 2 and Ω1 ( G) Z( G) or p = 2 and Ω2 ( G) Z( G). Also, we recall that a 2-group G is quasidihedral if it has an abelian maximal subgroup A of exponent > 2 and an involution t e G - A, which inverts each element in A.
In what follows G will be a 2-group of exponent > 4 satisfying the condition (*) .In Theorem 159.4 we determine up to isomorphism all such groups G which do not possess a normal elementary abelian subgroup of order 8. In Theorem 159.5 we classify our groups G with the assumption that Ü1(G) Z(G). It turns out that these groups must be 2-central, i.e., we must also have 02(G) Z(G). In Proposition 159.6 it is shown that if 02(G) is abelian, then G is also 2-central. Modular 2-groups (i.e., 2-groups that are D8-free) are classified in Theorem 159.7. Then we consider 2-groups G which do not have D8 as a subgroup or equivalently in which Ü1(G) is elementary abelian but 01(G) Z(G), and we show that such groups are of exponent 4 (Theorem 159.8). Finally, we classify in Theorem 159.9 our groups G in which Ü1(G) is nonabelian (i.e., D8 is a subgroup in G). The sta...

Table of contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. List of definitions and notations
  6. Preface
  7. § 145 p-groups all of whose maximal subgroups, except one, have derived subgroup of order ≤ p
  8. § 146 p-groups all of whose maximal subgroups, except one, have cyclic derived subgroups
  9. § 147 p-groups with exactly two sizes of conjugate classes
  10. § 148 Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic
  11. § 149 p-groups with many minimal nonabelian subgroups
  12. § 150 The exponents of finite p-groups and their automorphism groups
  13. § 151 p-groups all of whose nonabelian maximal subgroups have the largest possible center
  14. § 152 p-central p-groups
  15. § 153 Some generalizations of 2-central 2-groups
  16. § 154 Metacyclic p-groups covered by minimal nonabelian subgroups
  17. § 155 A new type of Thompson subgroup
  18. § 156 Minimal number of generators of a p-group, p > 2
  19. § 157 Some further properties of p-central p-groups
  20. § 158 On extraspecial normal subgroups of p-groups
  21. § 159 2-groups all of whose cyclic subgroups A, B with A ⋂ B ≠ {1} generate an abelian subgroup
  22. § 160 p-groups, p > 2, all of whose cyclic subgroups A, B with A ⋂ B ≠ {1} generate an abelian subgroup
  23. § 161 p-groups where all subgroups not contained in the Frattini subgroup are quasinormal
  24. § 162 The centralizer equality subgroup in a p-group
  25. § 163 Macdonald’s theorem on p-groups all of whose proper subgroups are of class at most 2
  26. § 164 Partitions and Hp-subgroups of a p-group
  27. § 165 p-groups G all of whose subgroups containing Φ(G) as a subgroup of index p are minimal nonabelian
  28. § 166 A characterization of p-groups of class > 2 all of whose proper subgroups are of class ≤ 2
  29. § 167 Nonabelian p-groups all of whose nonabelian subgroups contain the Frattini subgroup
  30. § 168 p-groups with given intersections of certain subgroups
  31. § 169 Nonabelian p-groups G with 〈A, B〉 minimal nonabelian for any two distinct maximal cyclic subgroups A, B of G
  32. § 170 p-groups with many minimal nonabelian subgroups, 2
  33. § 171 Characterizations of Dedekindian 2-groups
  34. § 172 On 2-groups with small centralizers of elements
  35. § 173 Nonabelian p-groups with exactly one noncyclic maximal abelian subgroup
  36. § 174 Classification of p-groups all of whose nonnormal subgroups are cyclic or abelian of type (p, p)
  37. § 175 Classification of p-groups all of whose nonnormal subgroups are cyclic, abelian of type (p, p) or ordinary quaternion
  38. § 176 Classification of p-groups with a cyclic intersection of any two distinct conjugate subgroups
  39. § 177 On the norm of a p-group
  40. § 178 p-groups whose character tables are strongly equivalent to character tables of metacyclic p-groups, and some related topics
  41. § 179 p-groups with the same numbers of subgroups of small indices and orders as in a metacyclic p-group
  42. § 180 p-groups all of whose noncyclic abelian subgroups are normal
  43. § 181 p-groups all of whose nonnormal abelian subgroups lie in the center of their normalizers
  44. § 182 p-groups with a special maximal cyclic subgroup
  45. § 183 p-groups generated by any two distinct maximal abelian subgroups
  46. § 184 p-groups in which the intersection of any two distinct conjugate subgroups is cyclic or generalized quaternion
  47. § 185 2-groups in which the intersection of any two distinct conjugate subgroups is either cyclic or of maximal class
  48. § 186 p-groups in which the intersection of any two distinct conjugate subgroups is either cyclic or abelian of type (p, p)
  49. § 187 p-groups in which the intersection of any two distinct conjugate cyclic subgroups is trivial
  50. § 188 p-groups with small subgroups generated by two conjugate elements
  51. § 189 2-groups with index of every cyclic subgroup in its normal closure ≤ 4
  52. Appendix 45 Varia II
  53. Appendix 46 On Zsigmondy primes
  54. Appendix 47 The holomorph of a cyclic 2-group
  55. Appendix 48 Some results of R. van der Waall and close to them
  56. Appendix 49 Kegel’s theorem on nilpotence of Hp-groups
  57. Appendix 50 Sufficient conditions for 2-nilpotence
  58. Appendix 51 Varia III
  59. Appendix 52 Normal complements for nilpotent Hall subgroups
  60. Appendix 53 p-groups with large abelian subgroups and some related results
  61. Appendix 54 On Passman’s Theorem 1.25 for p > 2
  62. Appendix 55 On p-groups with the cyclic derived subgroup of index p2
  63. Appendix 56 On finite groups all of whose p-subgroups of small orders are normal
  64. Appendix 57 p-groups with a 2-uniserial subgroup of order p and an abelian subgroup of type (p, p)
  65. Research problems and themes IV
  66. Bibliography
  67. Author index
  68. Subject index
  69. Endnotes

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Yes, you can access Groups of Prime Power Order. Volume 4 by Yakov G. Berkovich,Zvonimir Janko in PDF and/or ePUB format, as well as other popular books in Mathematics & Algebra. We have over 1.5 million books available in our catalogue for you to explore.