Groups of Prime Power Order. Volume 2This 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. |
Contents
| 1 | |
| 14 | |
| 19 | |
| 28 | |
| 43 | |
51 2groups with self centralizing subgroup isomorphic to E8 | 52 |
52 2groups with 2subgroup of small order | 75 |
53 2groups G with c2G 4 | 96 |
77 2groups with a selfcentralizing abelian subgroup of type 4 2 | 316 |
78 Minimal nonmodular pgroups | 323 |
79 Nonmodular quaternionfree 2groups | 334 |
80 Minimal nonquaternionfree 2groups | 356 |
81 Maximal abelian subgroups in 2groups | 361 |
82 A classification of 2groups with exactly three involutions | 368 |
83 pgroups G with Ω2G or Ω2G extraspecial | 396 |
84 2groups whose nonmetacyclic subgroups are generated by involutions | 399 |
54 2groups G with cnG 4 n 2 | 109 |
55 2groups G with small subgroup x G ox 2 | 122 |
56 Theorem of Ward on quaternionfree 2groups | 134 |
57 Nonabelian 2groups all of whose minimal nonabelian subgroups are isomorphic and have exponent 4 | 140 |
58 NonDedekindian pgroups all of whose nonnormal subgroups of the same order are conjugate | 147 |
59 pgroups with few nonnormal subgroups | 150 |
60 The structure of the Burnside group of order 212 | 151 |
61 Groups of exponent 4 generated by three involutions | 163 |
62 Groups with large normal closures of nonnormal cyclic subgroups | 169 |
63 Groups all of whose cyclic subgroups of composite orders are normal | 172 |
64 pgroups generated by elements of given order | 179 |
65 A2groups | 188 |
66 A new proof of Blackburns theorem on minimal nonmetacyclic 2groups | 197 |
67 Determination of U2groups | 202 |
68 Characterization of groups of prime exponent | 206 |
69 Elementary proofs of some Blackburns theorems | 209 |
70 Non2generator pgroups all of whose maximal subgroups are 2generator | 214 |
71 Determination of A2groups | 233 |
72 Angroups n 2 | 248 |
73 Classification of modular pgroups | 257 |
74 pgroups with a cyclic subgroup of index p2 | 274 |
75 Elements of order 4 in pgroups | 277 |
76 pgroups with few A1subgroups | 282 |
85 2groups with a nonabelian Frattini subgroup of order 16 | 402 |
86 pgroups G with metacyclic Ω2G | 406 |
87 2groups with exactly one nonmetacyclic maximal subgroup | 412 |
88 Hall chains in normal subgroups of pgroups | 437 |
89 2groups with exactly six cyclic subgroups of order 4 | 454 |
90 Nonabelian 2groups all of whose minimal nonabelian subgroups are of order 8 | 463 |
91 Maximal abelian subgroups of pgroups | 467 |
92 On minimal nonabelian subgroups of pgroups | 474 |
Appendix 16 Some central products | 485 |
Appendix 17 Alternate proofs of characterization theorems of Miller and Janko on 2groups and some related results | 492 |
Appendix 18 Replacement theorems | 501 |
Appendix 19 New proof of Wards theorem on quaternionfree 2groups | 506 |
Appendix 20 Some remarks on automorphisms | 509 |
Appendix 21 Isaacs examples | 512 |
Appendix 22 Minimal nonnilpotent groups | 516 |
Appendix 23 Groups all of whose noncentral conjugacy classes have the same size | 519 |
Appendix 24 On modular 2groups | 522 |
Appendix 25 Schreiers inequality for pgroups | 526 |
Appendix 26 pgroups all of whose nonabelian maximal subgroups are either absolutely regular or of maximal class | 529 |
Research problems and themes II | 531 |
| 569 | |
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Yakov Berkovich; Zvonimir Janko: Groups of Prime Power Order/Groups of Prime ... Yakov G. Berkovich,Zvonimir Janko No preview available - 2016 |