Yakov Berkovich; Zvonimir Janko: Groups of Prime Power Order. Volume 4 (De Gruyter Expositions in Mathematics Book 61)

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. Content List of definitions and notations Preface § 145 p-groups all of whose maximal subgroups, except one, have derived subgroup of order ≤ p § 146 p-groups all of whose maximal subgroups, except one, have cyclic derived subgroups § 147 p-groups with exactly two sizes of conjugate classes § 148 Maximal abelian and minimal nonabelian subgroups of some finite two-generator p-groups especially metacyclic § 149 p-groups with many minimal nonabelian subgroups § 150 The exponents of finite p-groups and their automorphism groups § 151 p-groups all of whose nonabelian maximal subgroups have the largest possible center § 152 p-central p-groups § 153 Some generalizations of 2-central 2-groups § 154 Metacyclic p-groups covered by minimal nonabelian subgroups § 155 A new type of Thompson subgroup § 156 Minimal number of generators of a p-group, p > 2 § 157 Some further properties of p-central p-groups § 158 On extraspecial normal subgroups of p-groups § 159 2-groups all of whose cyclic subgroups A, B with A ⋂ B ≠ {1} generate an abelian subgroup § 160 p-groups, p > 2, all of whose cyclic subgroups A, B with A ⋂ B ≠ {1} generate an abelian subgroup § 161 p-groups where all subgroups not contained in the Frattini subgroup are quasinormal § 162 The centralizer equality subgroup in a p-group § 163 Macdonald’s theorem on p-groups all of whose proper subgroups are of class at most 2 § 164 Partitions and Hp-subgroups of a p-group § 165 p-groups G all of whose subgroups containing ∅G) as a subgroup of index p are minimal nonabelian § 166 A characterization of p-groups of class > 2 all of whose proper subgroups are of class ≤ 2 § 167 Nonabelian p-groups all of whose nonabelian subgroups contain the Frattini subgroup § 168 p-groups with given intersections of certain subgroups § 169 Nonabelian p-groups G with minimal nonabelian for any two distinct maximal cyclic subgroups A, B of G § 170 p-groups with many minimal nonabelian subgroups, 2 § 171 Characterizations of Dedekindian 2-groups § 172 On 2-groups with small centralizers of elements § 173 Nonabelian p-groups with exactly one noncyclic maximal abelian subgroup § 174 Classification of p-groups all of whose nonnormal subgroups are cyclic or abelian of type (p, p) § 175 Classification of p-groups all of whose nonnormal subgroups are cyclic, abelian of type (p, p) or ordinary quaternion § 176 Classification of p-groups with a cyclic intersection of any two distinct conjugate subgroups § 177 On the norm of a p-group § 178 p-groups whose character tables are strongly equivalent to character tables of metacyclic p-groups, and some related topics § 179 p-groups with the same numbers of subgroups of small indices and orders as in a metacyclic p-group § 180 p-groups all of whose noncyclic abelian subgroups are normal § 181 p-groups all of whose nonnormal abelian subgroups lie in the center of their normalizers § 182 p-groups with a special maximal cyclic subgroup § 183 p-groups generated by any two distinct maximal abelian subgroups § 184 p-groups in which the intersection of any two distinct conjugate subgroups is cyclic or generalized quaternion § 185 2-groups in which the intersection of any two distinct conjugate subgroups is either cyclic or of maximal class § 186 p-groups in which the intersection of any two distinct conjugate subgroups is either cyclic or abelian of type (p, p) § 187 p-groups in which the intersection of any two distinct conjugate cyclic subgroups is trivial § 188 p-groups with small subgroups generated by two conjugate elements § 189 2-groups with index of every cyclic subgroup in its normal closure ≤ 4 Appendix 45 Varia II Appendix 46 On Zsigmondy primes Appendix 47 The holomorph of a cyclic 2-group Appendix 48 Some results of R. van der Waall and close to them Appendix 49 Kegel’s theorem on nilpotence of Hp-groups Appendix 50 Sufficient conditions for 2-nilpotence Appendix 51 Varia III Appendix 52 Normal complements for nilpotent Hall subgroups Appendix 53 p-groups with large abelian subgroups and some related results Appendix 54 On Passman’s Theorem 1.25 for p > 2 Appendix 55 On p-groups with the cyclic derived subgroup of index p2 Appendix 56 On finite groups all of whose p-subgroups of small orders are normal Appendix 57 p-groups with a 2-uniserial subgroup of order p and an abelian subgroup of type (p, p) Research problems and themes IV Bibliography Author index Subject index 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{u2019}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 This is theforth volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume includetheory of linear algebrasand Liealgebras. The book contains many dozens of original exercises (with difficult exercises being solved) and a list of about 900 research problems and themes.