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Format: 1 online resource (664p.)

ISBN: 9783110254488

Series Statement: De Gruyter Expositions in Mathematics ; 56

Content: 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.

Note: 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 -- , 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 , In English.

Additional Edition: ISBN 978-3-11-020717-0

Language: English

DOI: 10.1515/9783110254488

URL: https://doi.org/10.1515/9783110254488