Groups of Prime Power Order Volume 2 (de Gruyter Expositions in Mathematics, 47)

This is the second of three volumes devoted to elementary finite p-group theory. Similar to the first volume, hundreds of important results are analyzed and, in many cases, simplified. Important topics presented in this monograph include: (a) classification of p-groups all of whose cyclic subgroups of composite orders are normal, (b) classification of 2-groups with exactly three involutions, (c) two proofs of Ward's theorem on quaternion-free groups, (d) 2-groups with small centralizers of an involution, (e) classification of 2-groups with exactly four cyclic subgroups of order 2n > 2, (f) two new proofs of Blackburn's theorem on minimal nonmetacyclic groups, (g) classification of p-groups all of whose subgroups of index pÂ2 are abelian, (h) classification of 2-groups all of whose minimal nonabelian subgroups have order 8, (i) p-groups with cyclic subgroups of index pÂ2 are classified. This volume contains hundreds of original exercises (with all difficult exercises being solved) and an extended list of about 700 open problems. The book is based on Volume 1, and it is suitable for researchers and graduate students of mathematics with a modest background on algebra. Machine Generated Contents Note: [§]93. Nonabelian 2-groups All Of Whose Minimal Nonabelian Subgroups Are Meta-cyclic And Have Exponent 4 -- [§]94. Nonabelian 2-groups All Of Whose Minimal Nonabelian Subgroups Are Non-metacyclic And Have Exponent 4 -- [§]95. Nonabelian 2-groups Of Exponent 2e Which Have No Minimal Nonabelian Subgroups Of Exponent 2e -- [§]96. Groups With At Most Two Conjugate Classes Of Nonnormal Subgroups -- [§]97.p-groups In Which Some Subgroups Are Generated By Elements Of Order P -- [§]98. Nonabelian 2-groups All Of Whose Minimal Nonabelian Subgroups Are Isomorphic To M2n+1, N [≥] 3 Fixed -- [§]99.2-groups With Sectional Rank At Most 4 -- [§]100.2-groups With Exactly One Maximal Subgroup Which Is Neither Abelian Nor Minimal Nonabelian -- [§]101.p-groups G With P> 2 And D(g) = 2 Having Exactly One Maximal Subgroup Which Is Neither Abelian Nor Minimal Nonabelian. Note Continued: [§]102.p-groups G With P> 2 And D(g)> 2 Having Exactly One Maximal Subgroup Which Is Neither Abelian Nor Minimal Nonabelian -- [§]103. Some Results Of Jonah And Konvisser -- [§]104. Degrees Of Irreducible Characters Of P-groups Associated With Finite Algebras -- [§]105. On Some Special P-groups -- [§]106. On Maximal Subgroups Of Two-generator 2-groups -- [§]107. Ranks Of Maximal Subgroups Of Nonmetacyclic Two-generator 2-groups -- [§]108.p-groups With Few Conjugate Classes Of Minimal Nonabelian Subgroups -- [§]109. On P-groups With Metacyclic Maximal Subgroup Without Cyclic Subgroup Of Index P -- [§]110. Equilibrated P-groups -- [§]111. Characterization Of Abelian And Minimal Nonabelian Groups -- [§]112. Non-dedekindian P-groups All Of Whose Nonnormal Subgroups Have The Same Order -- [§]113. The Class Of 2-groups In [§]70 Is Not Bounded. Note Continued: [§]114. Further Counting Theorems -- [§]115. Finite P-groups All Of Whose Maximal Subgroups Except One Are Extraspecial -- [§]116. Groups Covered By Few Proper Subgroups -- [§]117.2-groups All Of Whose Nonnormal Subgroups Are Either Cyclic Or Of Maximal Class -- [§]118. Review Of Characterizations Of P-groups With Various Minimal Nonabelian Subgroups -- [§]119. Review Of Characterizations Of P-groups Of Maximal Class -- [§]120. Nonabelian 2-groups Such That Any Two Distinct Minimal Nonabelian Subgroups Have Cyclic Intersection -- [§]121.p-groups Of Breadth 2 -- [§]122.p-groups All Of Whose Subgroups Have Normalizers Of Index At Most P -- [§]123. Subgroups Of Finite Groups Generated By All Elements In Two Shortest Conjugacy Classes -- [§]124. The Number Of Subgroups Of Given Order In A Metacyclic P-group. Note Continued: [§]125.p-groups G Containing A Maximal Subgroup H All Of Whose Subgroups Are G-invariant -- [§]126. The Existence Of P-groups G1 2......Page 125 §55. 2-groups G with small subgroup (x ∈ G | o(x) = 2")......Page 138 §56. Theorem of Ward on quaternion-free 2-groups......Page 150 §57. Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic and have exponent 4......Page 156 §58. Non-Dedekindian p-groups all of whose nonnormal subgroups of the same order are conjugate......Page 163 §59. p-groups with few nonnormal subgroups......Page 166 §60. The structure of the Burnside group of order 212......Page 167 §61. Groups of exponent 4 generated by three involutions......Page 179 §62. Groups with large normal closures of nonnormal cyclic subgroups......Page 185 §63. Groups all of whose cyclic subgroups of composite orders are normal......Page 188 §64. p-groups generated by elements of given order......Page 195 §65. A2-groups......Page 204 §66. A new proof of Blackburn’s theorem on minimal nonmetacyclic 2-groups......Page 213 §67. Determination of U2-groups......Page 218 §68. Characterization of groups of prime exponent......Page 222 §69. Elementary proofs of some Blackburn’s theorems......Page 225 §70. Non-2-generator p-groups all of whose maximal subgroups are 2-generator......Page 230 §71. Determination of A2-groups......Page 249 §72. An-groups, n > 2......Page 264 §73. Classification of modular p-groups......Page 273 §74. p-groups with a cyclic subgroup of index p2......Page 290 §75. Elements of order ≤ 4 in p-groups......Page 293 §76. p-groups with few A1-subgroups......Page 298 §77. 2-groups with a self-centralizing abelian subgroup of type (4, 2)......Page 332 §78. Minimal nonmodular p-groups......Page 339 §79. Nonmodular quaternion-free 2-groups......Page 350 §80. Minimal non-quaternion-free 2-groups......Page 372 §81. Maximal abelian subgroups in 2-groups......Page 377 §82. A classification of 2-groups with exactly three involutions......Page 384 §83. p-groups G with Ω2(G) or Ω2*(G) extraspecial......Page 412 §84. 2-groups whose nonmetacyclic subgroups are generated by involutions......Page 415 §85. 2-groups with a nonabelian Frattini subgroup of order 16......Page 418 §86. p-groups G with metacyclic Ω2*(G)......Page 422 §87. 2-groups with exactly one nonmetacyclic maximal subgroup......Page 428 §88. Hall chains in normal subgroups of p-groups......Page 453 §89. 2-groups with exactly six cyclic subgroups of order 4......Page 470 §90. Nonabelian 2-groups all of whose minimal nonabelian subgroups are of order 8......Page 479 §91. Maximal abelian subgroups of p-groups......Page 483 §92. On minimal nonabelian subgroups of p-groups......Page 490 Appendix 16. Some central products......Page 501 Appendix 17. Alternate proofs of characterization theorems of Miller and Janko on 2-groups, and some related results......Page 508 Appendix 18. Replacement theorems......Page 517 Appendix 19. New proof of Ward’s theorem on quaternion-free 2-groups......Page 522 Appendix 20. Some remarks on automorphisms......Page 525 Appendix 21. Isaacs’ examples......Page 528 Appendix 22. Minimal nonnilpotent groups......Page 532 Appendix 23. Groups all of whose noncentral conjugacy classes have the same size......Page 535 Appendix 24. On modular 2-groups......Page 538 Appendix 25. Schreier’s inequality for p-groups......Page 542 Appendix 26. p-groups all of whose nonabelian maximal subgroups are either absolutely regular or of maximal class......Page 545 Research problems and themes II......Page 547 Backmatter ......Page 585

This is the second of three volumes devoted to elementary finite p-group theory. Similar to the first volume, hundreds of important results are analyzed and, in many cases, simplified. Important topics presented in this monograph include: (a) classification of p-groups all of whose cyclic subgroups of composite orders are normal, (b) classification of 2-groups with exactly three involutions, (c) two proofs of Ward's theorem on quaternion-free groups, (d) 2-groups with small centralizers of an involution, (e) classification of 2-groups with exactly four cyclic subgroups of order 2n > 2, (f) two new proofs of Blackburn's theorem on minimal nonmetacyclic groups, (g) classification of p-groups all of whose subgroups of index p2 are abelian, (h) classification of 2-groups all of whose minimal nonabelian subgroups have order 8, (i) p-groups with cyclic subgroups of index p2 are classified.

This volume contains hundreds of original exercises (with all difficult exercises being solved) and an extended list of about 700 open problems. The book is based on Volume 1, and it is suitable for researchers and graduate students of mathematics with a modest background on algebra.