Groups of Prime Power Order. / Volume 3 / / Yakov Berkovich, Zvonimir Janko.

• Superior document: Title is part of eBook package: De Gruyter DG Expositions in Mathematics Backlist eBook Package
• VerfasserIn: Berkovich, Yakov,; Janko, Zvonimir,
• Place / Publishing House: Berlin ;, Boston : : De Gruyter, , [2011]; ©2011
• Year of Publication: 2011
• Language: English
• Series: De Gruyter Expositions in Mathematics , 56
• Physical Description: 1 online resource (639 p.)

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.

Table of Contents

Frontmatter -- 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 -- A.27 Wreathed 2-groups -- A.28 Nilpotent subgroups -- A.29 Intersections of subgroups -- A.30 Thompson’s lemmas -- A.31 Nilpotent p'-subgroups of class 2 in GL(n, p) -- A.32 On abelian subgroups of given exponent and small index -- A.33 On Hadamard 2-groups -- A.34 Isaacs–Passman’s theorem on character degrees -- A.35 Groups of Frattini class 2 -- A.36 Hurwitz’ theorem on the composition of quadratic forms -- A.37 On generalized Dedekindian groups -- A.38 Some results of Blackburn and Macdonald -- A.39 Some consequences of Frobenius’ normal p-complement theorem -- A.40 Varia -- A.41 Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers -- A.42 On lattice isomorphisms of p-groups of maximal class -- A.43 Alternate proofs of two classical theorems on solvable groups and some related results -- A.44 Some of Freiman’s results on finite subsets of groups with small doubling -- Research problems and themes III -- Author index -- Subject index