Groups of Prime Power Order: Volume 3

Description

This is the third volume of a comprehensive and elementary treatment of finite p-group theory. Topics covered in this volume: impact of minimal nonabelian subgroups on the structure of p-groups, classification of groups all of whose nonnormal subgroups have the same order, degrees of irreducible characters of p-groups associated with finite algebras, groups covered by few proper subgroups, p-groups of element breadth 2 and subgroup breadth 1, exact number of subgroups of given order in a metacyclic p-group, soft subgroups, p-groups with a maximal elementary abelian subgroup of order p2, p-groups generated by certain minimal nonabelian subgroups, 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 about 900 research problems and themes. Contents List of definitions and notations Preface Prerequisites from Volumes 1 and 2 §93 Nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4 §94 Nonabelian 2-groups all of whose minimal nonabelian subgroups are nonmetacyclic 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 §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 §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 §125 p-groups G containing a maximal subgroup H all of whose subgroups are G-invariant §126 The existence of p-groups G1 < G such that Aut(G1) ≅ Aut(G) §127 On 2-groups containing a maximal elementary abelian subgroup of order 4 §128 The commutator subgroup of p-groups with the subgroup breadth 1 §129 On two-generator 2-groups with exactly one maximal subgroup which is not two-generator §130 Soft subgroups of p-groups §131 p-groups with a 2-uniserial subgroup of order p §132 On centralizers of elements in p-groups §133 Class and breadth of a p-group §134 On p-groups with maximal elementary abelian subgroup of order p2 §135 Finite p-groups generated by certain minimal nonabelian subgroups §136 p-groups in which certain proper nonabelian subgroups are two-generator §137 p-groups all of whose proper subgroups have its derived subgroup of order at most p §138 p-groups all of whose nonnormal subgroups have the smallest possible normalizer §139 p-groups with a noncyclic commutator group all of whose proper subgroups have a cyclic commutator group §140 Power automorphisms and the norm of a p-group §141 Nonabelian p-groups having exactly one maximal subgroup with a noncyclic center §142 Nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian §143 Alternate proof of the Reinhold Baer theorem on 2-groups with nonabelian norm §144 p-groups with small normal closures of all cyclic subgroups Appendix 27 Wreathed 2-groups Appendix 28 Nilpotent subgroups Appendix 29 Intersections of subgroups Appendix 30 Thompson’s lemmas Appendix 31 Nilpotent p’-subgroups of class 2 in GL(n, p) Appendix 32 On abelian subgroups of given exponent and small index Appendix 33 On Hadamard 2-groups Appendix 34 Isaacs–Passman’s theorem on character degrees Appendix 35 Groups of Frattini class 2 Appendix 36 Hurwitz’ theorem on the composition of quadratic forms Appendix 37 On generalized Dedekindian groups Appendix 38 Some results of Blackburn and Macdonald Appendix 39 Some consequences of Frobenius’ normal p-complement theorem Appendix 40 Varia Appendix 41 Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers Appendix 42 On lattice isomorphisms of p-groups of maximal class Appendix 43 Alternate proofs of two classical theorems on solvable groups and some related results Appendix 44 Some of Freiman’s results on finite subsets of groups with small doubling Research problems and themes III Author index Subject index