E-Book, Englisch, Band 56, 664 Seiten
Volume 3
E-Book, Englisch, Band 56, 664 Seiten
Reihe: De Gruyter Expositions in Mathematics
ISBN: 978-3-11-025448-8
Verlag: De Gruyter
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)
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.
Zielgruppe
Researchers, Graduate Students of Mathematics; Academic Libraries
Autoren/Hrsg.
Weitere Infos & Material
1;Contents;6
2;List of definitions and notations;10
3;Preface;16
4;Prerequisites from Volumes 1 and 2;18
5;§93 Nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4;28
6;§94 Nonabelian 2-groups all of whose minimal nonabelian subgroups are nonmetacyclic and have exponent 4;35
7;§95 Nonabelian 2-groups of exponent 2e which have no minimal nonabelian subgroups of exponent 2e;37
8;§96 Groups with at most two conjugate classes of nonnormal subgroups;39
9;§97 p-groups in which some subgroups are generated by elements of order p;51
10;§98 Nonabelian 2-groups all of whose minimal nonabelian subgroups are isomorphic to M2n+1, n> 3 fixed;58
11;§99 2-groups with sectional rank at most 4;61
12;§100 2-groups with exactly one maximal subgroup which is neither abelian nor minimal nonabelian;73
13;§101 p-groups G with p > 2 and d(G) = 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian;93
14;§102 p-groups G with p > 2 and d(G) > 2 having exactly one maximal subgroup which is neither abelian nor minimal nonabelian;104
15;§103 Some results of Jonah and Konvisser;120
16;§104 Degrees of irreducible characters of p-groups associated with finite algebras;124
17;§105 On some special p-groups;129
18;§106 On maximal subgroups of two-generator 2-groups;137
19;§107 Ranks of maximal subgroups of nonmetacyclic two-generator 2-groups;140
20;§108 p-groups with few conjugate classes of minimal nonabelian subgroups;147
21;§109 On p-groups with metacyclic maximal subgroup without cyclic subgroup of index p;149
22;§110 Equilibrated p-groups;152
23;§111 Characterization of abelian and minimal nonabelian groups;161
24;§112 Non-Dedekindian p-groups all of whose nonnormal subgroups have the same order;167
25;§113 The class of 2-groups in §70 is not bounded;175
26;§114 Further counting theorems;179
27;§115 Finite p-groups all of whose maximal subgroups except one are extraspecial;184
28;§116 Groups covered by few proper subgroups;189
29;§117 2-groups all of whose nonnormal subgroups are either cyclic or of maximal class;203
30;§118 Review of characterizations of p-groups with various minimal nonabelian subgroups;206
31;§119 Review of characterizations of p-groups of maximal class;212
32;§120 Nonabelian 2-groups such that any two distinct minimal nonabelian subgroups have cyclic intersection;219
33;§121 p-groups of breadth 2;224
34;§122 p-groups all of whose subgroups have normalizers of index at most p;231
35;§123 Subgroups of finite groups generated by all elements in two shortest conjugacy classes;264
36;§124 The number of subgroups of given order in a metacyclic p-group;266
37;§125 p-groups G containing a maximal subgroup H all of whose subgroups are G-invariant;296
38;§126 The existence of p-groups G1 < G such that Aut(G1) Aut(G);299
39;§127 On 2-groups containing a maximal elementary abelian subgroup of order 4;302
40;§128 The commutator subgroup of p-groups with the subgroup breadth 1;304
41;§129 On two-generator 2-groups with exactly one maximal subgroup which is not two-generator;312
42;§130 Soft subgroups of p-groups;314
43;§131 p-groups with a 2-uniserial subgroup of order p;319
44;§132 On centralizers of elements in p-groups;322
45;§133 Class and breadth of a p-group;327
46;§134 On p-groups with maximal elementary abelian subgroup of order p2;331
47;§135 Finite p-groups generated by certain minimal nonabelian subgroups;342
48;§136 p-groups in which certain proper nonabelian subgroups are two-generator;355
49;§137 p-groups all of whose proper subgroups have its derived subgroup of order at most p;365
50;§138 p-groups all of whose nonnormal subgroups have the smallest possible normalizer;370
51;§139 p-groups with a noncyclic commutator group all of whose proper subgroups have a cyclic commutator group;382
52;§140 Power automorphisms and the norm of a p-group;390
53;§141 Nonabelian p-groups having exactly one maximal subgroup with a noncyclic center;395
54;§142 Nonabelian p-groups all of whose nonabelian maximal subgroups are either metacyclic or minimal nonabelian;397
55;§143 Alternate proof of the Reinhold Baer theorem on 2-groups with nonabelian norm;400
56;§144 p-groups with small normal closures of all cyclic subgroups;403
57;Appendix 27 Wreathed 2-groups;411
58;Appendix 28 Nilpotent subgroups;420
59;Appendix 29 Intersections of subgroups;432
60;Appendix 30 Thompson’s lemmas;443
61;Appendix 31 Nilpotent p’-subgroups of class 2 in GL(n, p);455
62;Appendix 32 On abelian subgroups of given exponent and small index;461
63;Appendix 33 On Hadamard 2-groups;464
64;Appendix 34 Isaacs–Passman’s theorem on character degrees;467
65;Appendix 35 Groups of Frattini class 2;473
66;Appendix 36 Hurwitz’ theorem on the composition of quadratic forms;476
67;Appendix 37 On generalized Dedekindian groups;479
68;Appendix 38 Some results of Blackburn and Macdonald;484
69;Appendix 39 Some consequences of Frobenius’ normal p-complement theorem;487
70;Appendix 40 Varia;499
71;Appendix 41 Nonabelian 2-groups all of whose minimal nonabelian subgroups have cyclic centralizers;541
72;Appendix 42 On lattice isomorphisms of p-groups of maximal class;543
73;Appendix 43 Alternate proofs of two classical theorems on solvable groups and some related results;546
74;Appendix 44 Some of Freiman’s results on finite subsets of groups with small doubling;554
75;Research problems and themes III;563
76;Author index;657
77;Subject index;659
