UID:
edocfu_9959239253702883
Format:
1 online resource (668 p.)
Edition:
1st ed.
ISBN:
1-283-40037-5
,
9786613400376
,
3-11-025448-4
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:
Includes indexes.
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Frontmatter --
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Contents --
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List of definitions and notations --
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Preface --
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Prerequisites from Volumes 1 and 2 --
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§93 Nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4 --
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§94 Nonabelian 2-groups all of whose minimal nonabelian subgroups are nonmetacyclic and have exponent 4 --
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§95 Nonabelian 2-groups of exponent 2e which have no minimal nonabelian subgroups of exponent 2e --
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§96 Groups with at most two conjugate classes of nonnormal subgroups --
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§97 p-groups in which some subgroups are generated by elements of order p --
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§98 Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic to M2n+1, n 3 fixed --
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§99 2-groups with sectional rank at most 4 --
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§100 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian --
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§101 p-groups G with p 〉 2 and d(G) = 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian --
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§102 p-groups G with p 〉 2 and d(G) 〉 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian --
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§103 Some results of Jonah and Konvisser --
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§104 Degrees of irreducible characters of p-groups associated with finite algebras --
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§105 On some special p-groups --
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§106 On maximal subgroups of two-generator 2-groups --
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§107 Ranks of maximal subgroups of nonmetacyclic two-generator 2-groups --
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§108 p-groups with few conjugate classes of minimal nonabelian subgroups --
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§109 On p-groups with metacyclic maximal subgroup without cyclic subgroup of index p --
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§110 Equilibrated p-groups --
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§111 Characterization of abelian and minimal nonabelian groups --
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§112 Non-Dedekindian p-groups all of whose nonnormal subgroups have the same order --
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§113 The class of 2-groups in §70 is not bounded --
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§114 Further counting theorems --
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§115 Finite p-groups all of whose maximal subgroups except one are extraspecial --
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§116 Groups covered by few proper subgroups --
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§117 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class --
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§118 Review of characterizations of p-groups with various minimal nonabelian subgroups --
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§119 Review of characterizations of p-groups of maximal class --
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§120 Nonabelian 2-groups such that any two distinct minimal nonabelian subgroups have cyclic intersection --
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§121 p-groups of breadth 2 --
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§122 p-groups all of whose subgroups have normalizers of index at most p --
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§123 Subgroups of finite groups generated by all elements in two shortest conjugacy classes --
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§124 The number of subgroups of given order in a metacyclic p-group --
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§125 p-groups G containing a maximal subgroup H all of whose subgroups are G-invariant --
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§126 The existence of p-groups G1 G such that Aut(G1) Aut(G) --
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§127 On 2-groups containing a maximal elementary abelian subgroup of order 4 --
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§128 The commutator subgroup of p-groups with the subgroup breadth 1 --
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§129 On two-generator 2-groups with exactly one maximal subgroup which is not two-generator --
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§130 Soft subgroups of p-groups --
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§131 p-groups with a 2-uniserial subgroup of order p --
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§132 On centralizers of elements in p-groups --
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§133 Class and breadth of a p-group --
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§134 On p-groups with maximal elementary abelian subgroup of order p2 --
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§135 Finite p-groups generated by certain minimal nonabelian subgroups --
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§136 p-groups in which certain proper nonabelian subgroups are two-generator --
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§137 p-groups all of whose proper subgroups have its derived subgroup of order at most p --
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§138 p-groups all of whose nonnormal subgroups have the smallest possible normalizer --
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§139 p-groups with a noncyclic commutator group all of whose proper subgroups have a cyclic commutator group --
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§140 Power automorphisms and the norm of a p-group --
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§141 Nonabelian p-groups having exactly one maximal subgroup with a noncyclic center --
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§142 Nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian --
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§143 Alternate proof of the Reinhold Baer theorem on 2-groups with nonabelian norm --
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§144 p-groups with small normal closures of all cyclic subgroups --
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Appendix 27 Wreathed 2-groups --
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Appendix 28 Nilpotent subgroups --
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Appendix 29 Intersections of subgroups --
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Appendix 30 Thompson's lemmas --
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Appendix 31 Nilpotent p'-subgroups of class 2 in GL(n, p) --
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Appendix 32 On abelian subgroups of given exponent and small index --
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Appendix 33 On Hadamard 2-groups --
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Appendix 34 Isaacs-Passman's theorem on character degrees --
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Appendix 35 Groups of Frattini class 2 --
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Appendix 36 Hurwitz' theorem on the composition of quadratic forms --
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Appendix 37 On generalized Dedekindian groups --
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Appendix 38 Some results of Blackburn and Macdonald --
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Appendix 39 Some consequences of Frobenius' normal p-complement theorem --
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Appendix 40 Varia --
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Appendix 41 Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers --
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Appendix 42 On lattice isomorphisms of p-groups of maximal class --
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Appendix 43 Alternate proofs of two classical theorems on solvable groups and some related results --
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Appendix 44 Some of Freiman's results on finite subsets of groups with small doubling --
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Research problems and themes III --
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Author index --
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Subject index
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Issued also in print.
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English
Additional Edition:
ISBN 3-11-020717-6
Language:
English
DOI:
10.1515/9783110254488