## Yakov Berkovich; Zvonimir Janko: Groups of Prime Power Order, Volume 1This 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

1 | |

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 |

480 | |

### Other editions - View all

### Common terms and phrases

Ä pk Ä Z.G abelian of type abelian p-group absolutely regular assume that G Aut.G Aut(G automorphism characteristic subgroup characters of G cl(G class and order Classify the p-groups conjugacy classes contradiction Corollary cyclic of order divides elements of order epimorphic Exercise exp.G exp(G extraspecial G contains G is irregular G of order G-invariant subgroup group G group of order Hence homocyclic index p2 induction on jGj involutions Irr.G irreducible characters irregular p-groups jG0j Lemma Let G Let H maximal class maximal subgroups minimal normal subgroup nilpotent nonabelian of order nonabelian p-group noncyclic nonmetacyclic normal in G ofmaximal order p3 p-group G p-group of maximal p3 and exponent powerful p-group Proposition prove semidirect product Study the p-groups subgroup H subgroup of G subgroup of index subgroups of order Suppose that G Sylow p-subgroup Theorem Theorem 1.2 wreath product