## Yakov Berkovich; Zvonimir Janko: Groups of Prime Power Order, Volume 2This is the second of three volumes on finite p-group theory, written by two prominent authors in the area. |

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### Contents

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51 2groups with self centralizing subgroup isomorphic to E8 | 52 |

52 2groups with 2subgroup of small order | 75 |

53 2groups G with c2G 4 | 96 |

77 2groups with a selfcentralizing abelian subgroup of type 4 2 | 316 |

78 Minimal nonmodular pgroups | 323 |

79 Nonmodular quaternionfree 2groups | 334 |

80 Minimal nonquaternionfree 2groups | 356 |

81 Maximal abelian subgroups in 2groups | 361 |

82 A classification of 2groups with exactly three involutions | 368 |

83 pgroups G with Ω2G or Ω2G extraspecial | 396 |

84 2groups whose nonmetacyclic subgroups are generated by involutions | 399 |

54 2groups G with cnG 4 n 2 | 109 |

55 2groups G with small subgroup x G ox 2 | 122 |

56 Theorem of Ward on quaternionfree 2groups | 134 |

57 Nonabelian 2groups all of whose minimal nonabelian subgroups are isomorphic and have exponent 4 | 140 |

58 NonDedekindian pgroups all of whose nonnormal subgroups of the same order are conjugate | 147 |

59 pgroups with few nonnormal subgroups | 150 |

60 The structure of the Burnside group of order 212 | 151 |

61 Groups of exponent 4 generated by three involutions | 163 |

62 Groups with large normal closures of nonnormal cyclic subgroups | 169 |

63 Groups all of whose cyclic subgroups of composite orders are normal | 172 |

64 pgroups generated by elements of given order | 179 |

65 A2groups | 188 |

66 A new proof of Blackburns theorem on minimal nonmetacyclic 2groups | 197 |

67 Determination of U2groups | 202 |

68 Characterization of groups of prime exponent | 206 |

69 Elementary proofs of some Blackburns theorems | 209 |

70 Non2generator pgroups all of whose maximal subgroups are 2generator | 214 |

71 Determination of A2groups | 233 |

72 Angroups n 2 | 248 |

73 Classification of modular pgroups | 257 |

74 pgroups with a cyclic subgroup of index p2 | 274 |

75 Elements of order 4 in pgroups | 277 |

76 pgroups with few A1subgroups | 282 |

85 2groups with a nonabelian Frattini subgroup of order 16 | 402 |

86 pgroups G with metacyclic Ω2G | 406 |

87 2groups with exactly one nonmetacyclic maximal subgroup | 412 |

88 Hall chains in normal subgroups of pgroups | 437 |

89 2groups with exactly six cyclic subgroups of order 4 | 454 |

90 Nonabelian 2groups all of whose minimal nonabelian subgroups are of order 8 | 463 |

91 Maximal abelian subgroups of pgroups | 467 |

92 On minimal nonabelian subgroups of pgroups | 474 |

Appendix 16 Some central products | 485 |

Appendix 17 Alternate proofs of characterization theorems of Miller and Janko on 2groups and some related results | 492 |

Appendix 18 Replacement theorems | 501 |

Appendix 19 New proof of Wards theorem on quaternionfree 2groups | 506 |

Appendix 20 Some remarks on automorphisms | 509 |

Appendix 21 Isaacs examples | 512 |

Appendix 22 Minimal nonnilpotent groups | 516 |

Appendix 23 Groups all of whose noncentral conjugacy classes have the same size | 519 |

Appendix 24 On modular 2groups | 522 |

Appendix 25 Schreiers inequality for pgroups | 526 |

Appendix 26 pgroups all of whose nonabelian maximal subgroups are either absolutely regular or of maximal class | 529 |

Research problems and themes II | 531 |

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### Other editions - View all

Yakov Berkovich; Zvonimir Janko: Groups of Prime Power Order, Volume 1 Yakov Berkovich Limited preview - 2008 |

### Common terms and phrases

A1-group A1-subgroups abelian maximal subgroup abelian of type assume that G C4 X C2 carmot centralizes characteristic subgroup Classify the p-groups conjugacy classes conjugate contradiction cyclic of order cyclic subgroup dihedral elementary abelian subgroup elements of order exp(G G containing G is metacyclic G of order G Z(G G-invariant subgroup group G group of order Hence Hp-chain implies induces involution isomorphic jG0j Lemma Let G maximal abelian subgroup maximal class metacyclic group minimal nonabelian subgroup modular nonabelian of order nonabelian p-group noncyclic nonnormal normal abelian subgroup normal elementary abelian normal four-subgroup normal in G oforder order 24 order p3 p-groups G Proof Proposition proved Q8-free quaternion quotient group Š C4 Š D8 Š E4 Š Q8 semidirect product set F1 Study the p-groups subgroup of G subgroup of index subgroup of order Suppose that G Sylow 2-subgroup Theorem wl1ich