Yakov Berkovich; Zvonimir Janko: Groups of Prime Power Order, Volume 1

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Walter de Gruyter, Dec 10, 2008 - Mathematics - 532 pages
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This is the first of three volumes on finite p-group theory. It presents the state of the art and in addition contains numerous new and easy proofs of famous theorems, many exercises (some of them with solutions), and about 1500 open problems. It is expected to be useful to certain applied mathematics areas, such as combinatorics, coding theory, and computer sciences. The book should also be easily comprehensible to students and scientists with some basic knowledge of group theory and algebra.

 

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Contents

Introduction
1
1 Groups with a cyclic subgroup of index p Frattini subgroup Varia
22
2 The class number character degrees
58
3 Minimal classes
69
4 pgroups with cyclic Frattini subgroup
73
5 Halls enumeration principle
81
6 qautomorphisms of qgroups
91
7 Regular pgroups
98
32 W Gaschützs and P Schmids theorems on pautomorphisms of pgroups
309
33 Groups of order pm with automorphisms of order pm1 pm2 or pm3
314
34 Nilpotent groups of automorphisms
318
35 Maximal abelian subgroups of pgroups
326
36 Short proofs of some basic characterization theorems of finite pgroup theory
333
37 MacWilliams theorem
345
38 pgroups with exactly two conjugate classes of subgroups of small orders and exponentp 2
348
39 Alperins problem on abelian subgroups of small index
351

8 Pyramidal pgroups
109
9 On pgroups of maximal class
114
10 On abelian subgroups of pgroups
128
11 On the power structure of a pgroup
146
12 Counting theorems for pgroups of maximal class
151
13 Further counting theorems
161
14 Thompsons critical subgroup
185
15 Generators of pgroups
189
16 Classification of finite pgroups all of whose noncyclic subgroups are normal
192
17 Counting theorems for regular pgroups
198
18 Counting theorems for irregular pgroups
202
19 Some additional counting theorems
215
20 Groups with small abelian subgroups and partitions
219
21 On the Schur multiplier and the commutator subgroup
222
22 On characters of pgroups
229
23 On subgroups of given exponent
242
24 Halls theorem on normal subgroups of given exponent
246
25 On the lattice of subgroups of a group
256
26 Powerful pgroups
262
27 pgroups with normal centralizers of all elements
275
28 pgroups with a uniqueness condition for nonnormal subgroups
279
29 On isoclinism
285
30 On pgroups with few nonabelian subgroups of order pp and exponent p
289
31 On pgroups with small p0groups of operators
301
40 On breadth and class number of pgroups
355
41 Groups in which every two noncyclic subgroups of the same order have the same rank
358
42 On intersections of some subgroups
362
43 On 2groups with few cyclic subgroups of given order
365
44 Some characterizations of metacyclic pgroups
372
45 A counting theorem for pgroups of odd order
377
Appendix 1 The HallPetrescu formula
379
Appendix 2 Manns proof of monomiality of pgroups
383
Appendix 3 Theorems of Isaacs on actions of groups
385
Appendix 4 Freimans numbertheoretical theorems
393
Appendix 5 Another proof of Theorem 54
399
Appendix 6 On the order of pgroups of given derived length
401
Appendix 7 Relative indices of elements of pgroups
405
Appendix 8 pgroups withabsolutely regular Frattini subgroup
409
Appendix 9 On characteristic subgroups of metacyclic groups
412
Appendix 10 On minimal characters of pgroups
417
Appendix 11 On sums of degrees of irreducible characters
419
Appendix 12 2groups whose maximal cyclic subgroups of order 2 are selfcentralizing
422
Appendix 13 Normalizers of Sylow psubgroups of symmetric groups
425
Appendix 14 2groups with an involution contained in only one subgroup of order 4
431
Appendix 15 A criterion for a group to be nilpotent
433
Research problems and themes I
437
Backmatter
480
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About the author (2008)

Yakov Berkovich , University of Haifa, Israel.

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