Groups of Prime Power Order, Volume 3 (De Gruyter Expositions in Mathematics, 56)

This is the last of three volumes of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume: (a) impact of minimal nonabelian subgroups on the structure of p-groups, (b) classification of groups all of whose nonnormal subgroups have the same order, (c) degrees of irreducible characters of p-groups associated with finite algebras, (d) groups covered by few proper subgroups, (e) p-groups of element breadth 2 and subgroup breadth 1, (f) exact number of subgroups of given order in a metacyclic p-group, (g) soft subgroups, (h) p-groups with a maximal elementary abelian subgroup of order p??, (i) p-groups generated by certain minimal nonabelian subgroups, (j) p-groups in which certain nonabelian subgroups are 2-generator. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of more than 1000 research problems and themes. Groups of Prime Power Order - Volume 3 ......Page 1 De Gruyter Expositions in Mathematics 56......Page 2 ISBN: 9783110207170......Page 5 --> Contents......Page 6 List of definitions and notations......Page 10 Preface......Page 16 Prerequisites from Volumes 1 and 2......Page 18 § 93 - Nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4......Page 28 § 94 - Nonabelian 2-groups all of whose minimal nonabelian subgroups are nonmetacyclic and have exponent 4......Page 35 § 95 - Nonabelian 2-groups of exponent 2e which have no minimal nonabelian subgroups of exponent 2e......Page 37 § 96 - Groups with at most two conjugate classes of nonnormal subgroups......Page 39 § 97 - p-groups in which some subgroups are generated by elements of order p......Page 51 § 98 - Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic to M2nC1, n > 3 fixed......Page 58 § 99 - 2-groups with sectional rank at most 4......Page 61 § 100 - 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian......Page 73 § 102 - p-groups G with p > 2 and d. (G) > 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian......Page 93 § 101 - p-groups G with p > 2 and d (G) = 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian......Page 109 § 103 - Some results of Jonah and Konvisser......Page 120 § 104 - Degrees of irreducible characters of p-groups associated with finite algebras......Page 124 § 105 - On some special p-groups......Page 129 § 106 - On maximal subgroups of two-generator 2-groups......Page 137 § 107 - Ranks of maximal subgroups of nonmetacyclic two-generator 2-groups......Page 140 § 108 - p-groups with few conjugate classes of minimal nonabelian subgroups......Page 147 § 109 - On p-groups with metacyclic maximal subgroup without cyclic subgroup of index p......Page 149 § 110 - Equilibrated p-groups......Page 152 § 111 - Characterization of abelian and minimal nonabelian groups......Page 161 § 112 - Non-Dedekindian p-groups all of whose nonnormal subgroups have the same order......Page 167 § 113 - The class of 2-groups in § 70 is not bounded......Page 175 § 114 - Further counting theorems......Page 179 § 115 - Finite p-groups all of whose maximal subgroups except one are extraspecial......Page 184 § 116 - Groups covered by few proper subgroups......Page 189 § 117 - 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class......Page 203 § 118 - Review of characterizations of p-groups with various minimal nonabelian subgroups......Page 206 § 119 - Review of characterizations of p-groups of maximal class......Page 212 § 120 - Nonabelian 2-groups such that any two distinct minimal nonabelian subgroups have cyclic intersection......Page 219 § 121 - p-groups of breadth 2......Page 224 § 122 - p-groups all of whose subgroups have normalizers of index at most p......Page 231 § 123 - Subgroups of finite groups generated by all elements in two shortest conjugacy classes......Page 264 § 124 - The number of subgroups of given order in a metacyclic p-group......Page 266 § 125 - p-groups G containing a maximal subgroup H all of whose subgroups are G-invariant......Page 296 § 126 - The existence of p-groups G1 < G such that Aut (G1) = Aut (G)......Page 299 § 127 - On 2-groups containing a maximal elementary abelian subgroup of order 4......Page 302 § 128 - The commutator subgroup of p-groups with the subgroup breadth 1......Page 304 § 129 - On two-generator 2-groups with exactly one maximal subgroup which is not two-generator......Page 312 § 130 - Soft subgroups of p-groups......Page 314 § 131 - p-groups with a 2-uniserial subgroup of order p......Page 319 § 132 - On centralizers of elements in p-groups......Page 322 § 133 - Class and breadth of a p-group......Page 327 § 134 - On p-groups with maximal elementary abelian subgroup of order p2......Page 331 § 135 - Finite p-groups generated by certain minimal nonabelian subgroups......Page 342 § 136 - p-groups in which certain proper nonabelian subgroups are two-generator......Page 355 § 137 - p-groups all of whose proper subgroups have its derived subgroup of order at most p......Page 365 § 138 - p-groups all of whose nonnormal subgroups have the smallest possible normalizer......Page 370 § 139 - p-groups with a noncyclic commutator group all of whose proper subgroups have a cyclic commutator group......Page 382 § 140 - Power automorphisms and the norm of a p-group......Page 390 § 141 - Nonabelian p-groups having exactly one maximal subgroup with a noncyclic center......Page 395 § 142 - Nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian......Page 397 § 143 - Alternate proof of the Reinhold Baer theorem on 2-groups with nonabelian norm......Page 400 § 144 - p-groups with small normal closures of all cyclic subgroups......Page 403 Appendix 27 - Wreathed 2-groups......Page 411 Appendix 28 - Nilpotent subgroups......Page 420 Appendix 29 - Intersections of subgroups......Page 432 Appendix 30 - Thompson’s lemmas......Page 443 Appendix 31 - Nilpotent p'-subgroups of class 2 in GL (n; p)......Page 455 Appendix 32 - On abelian subgroups of given exponent and small index......Page 461 Appendix 33 - On Hadamard 2-groups......Page 464 Appendix 34 - Isaacs–Passman’s theorem on character degrees......Page 467 Appendix 35 - Groups of Frattini class 2......Page 473 Appendix 36 - Hurwitz’ theorem on the composition of quadratic forms......Page 476 Appendix 37 - On generalized Dedekindian groups......Page 479 Appendix 38 - Some results of Blackburn and Macdonald......Page 484 Appendix 39 - Some consequences of Frobenius’ normal p-complement theorem......Page 487 Appendix 40 - Varia......Page 499 Appendix 41 - Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers......Page 541 Appendix 42 - On lattice isomorphisms of p-groups of maximal class......Page 543 Appendix 43 - Alternate proofs of two classical theorems on solvable groups and some related results......Page 546 Appendix 44 - Some of Freiman’s results on finite subsets of groups with small doubling......Page 554 Research problems and themes III......Page 563 Bibliography......Page 620 Author index......Page 657 Subject index......Page 659 Groups of Prime Power Order - Volume 3 1 De Gruyter Expositions in Mathematics 56 2 ISBN: 9783110207170 5 --> Contents 6 List of definitions and notations 10 Preface 16 Prerequisites from Volumes 1 and 2 18 Chapters 28 § 93 - Nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4 28 § 94 - Nonabelian 2-groups all of whose minimal nonabelian subgroups are nonmetacyclic and have exponent 4 35 § 95 - Nonabelian 2-groups of exponent 2e which have no minimal nonabelian subgroups of exponent 2e 37 § 96 - Groups with at most two conjugate classes of nonnormal subgroups 39 § 97 - p-groups in which some subgroups are generated by elements of order p 51 § 98 - Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic to M2nC1, n > 3 fixed 58 § 99 - 2-groups with sectional rank at most 4 61 § 100 - 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian 73 § 102 - p-groups G with p > 2 and d. (G) > 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian 93 § 101 - p-groups G with p > 2 and d (G) = 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian 109 § 103 - Some results of Jonah and Konvisser 120 § 104 - Degrees of irreducible characters of p-groups associated with finite algebras 124 § 105 - On some special p-groups 129 § 106 - On maximal subgroups of two-generator 2-groups 137 § 107 - Ranks of maximal subgroups of nonmetacyclic two-generator 2-groups 140 § 108 - p-groups with few conjugate classes of minimal nonabelian subgroups 147 § 109 - On p-groups with metacyclic maximal subgroup without cyclic subgroup of index p 149 § 110 - Equilibrated p-groups 152 § 111 - Characterization of abelian and minimal nonabelian groups 161 § 112 - Non-Dedekindian p-groups all of whose nonnormal subgroups have the same order 167 § 113 - The class of 2-groups in § 70 is not bounded 175 § 114 - Further counting theorems 179 § 115 - Finite p-groups all of whose maximal subgroups except one are extraspecial 184 § 116 - Groups covered by few proper subgroups 189 § 117 - 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class 203 § 118 - Review of characterizations of p-groups with various minimal nonabelian subgroups 206 § 119 - Review of characterizations of p-groups of maximal class 212 § 120 - Nonabelian 2-groups such that any two distinct minimal nonabelian subgroups have cyclic intersection 219 § 121 - p-groups of breadth 2 224 § 122 - p-groups all of whose subgroups have normalizers of index at most p 231 § 123 - Subgroups of finite groups generated by all elements in two shortest conjugacy classes 264 § 124 - The number of subgroups of given order in a metacyclic p-group 266 § 125 - p-groups G containing a maximal subgroup H all of whose subgroups are G-invariant 296 § 126 - The existence of p-groups G1 < G such that Aut (G1) = Aut (G) 299 § 127 - On 2-groups containing a maximal elementary abelian subgroup of order 4 302 § 128 - The commutator subgroup of p-groups with the subgroup breadth 1 304 § 129 - On two-generator 2-groups with exactly one maximal subgroup which is not two-generator 312 § 130 - Soft subgroups of p-groups 314 § 131 - p-groups with a 2-uniserial subgroup of order p 319 § 132 - On centralizers of elements in p-groups 322 § 133 - Class and breadth of a p-group 327 § 134 - On p-groups with maximal elementary abelian subgroup of order p2 331 § 135 - Finite p-groups generated by certain minimal nonabelian subgroups 342 § 136 - p-groups in which certain proper nonabelian subgroups are two-generator 355 § 137 - p-groups all of whose proper subgroups have its derived subgroup of order at most p 365 § 138 - p-groups all of whose nonnormal subgroups have the smallest possible normalizer 370 § 139 - p-groups with a noncyclic commutator group all of whose proper subgroups have a cyclic commutator group 382 § 140 - Power automorphisms and the norm of a p-group 390 § 141 - Nonabelian p-groups having exactly one maximal subgroup with a noncyclic center 395 § 142 - Nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian 397 § 143 - Alternate proof of the Reinhold Baer theorem on 2-groups with nonabelian norm 400 § 144 - p-groups with small normal closures of all cyclic subgroups 403 Appendix 411 Appendix 27 - Wreathed 2-groups 411 Appendix 28 - Nilpotent subgroups 420 Appendix 29 - Intersections of subgroups 432 Appendix 30 - Thompson’s lemmas 443 Appendix 31 - Nilpotent p'-subgroups of class 2 in GL (n; p) 455 Appendix 32 - On abelian subgroups of given exponent and small index 461 Appendix 33 - On Hadamard 2-groups 464 Appendix 34 - Isaacs–Passman’s theorem on character degrees 467 Appendix 35 - Groups of Frattini class 2 473 Appendix 36 - Hurwitz’ theorem on the composition of quadratic forms 476 Appendix 37 - On generalized Dedekindian groups 479 Appendix 38 - Some results of Blackburn and Macdonald 484 Appendix 39 - Some consequences of Frobenius’ normal p-complement theorem 487 Appendix 40 - Varia 499 Appendix 41 - Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers 541 Appendix 42 - On lattice isomorphisms of p-groups of maximal class 543 Appendix 43 - Alternate proofs of two classical theorems on solvable groups and some related results 546 Appendix 44 - Some of Freiman’s results on finite subsets of groups with small doubling 554 Research problems and themes III 563 Bibliography 620 Author index 657 Subject index 659 3110207176,9783110207170 De Gruyter 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