Some unsolvable conjectures in finite p-groups

Abstract

We survey some unsolvable conjectures in finite p-groups and their research progress.

Key words： Finite p-groups; Hua-Tuan’s conjecture; Higman’s conjecture; Oliver’s conjecture; Wiegold’s conjecture

References

1 Abdollahi A . Finite p-groups of class 2 have noninner automorphisms of order p. J Algebra, 2007, 312 (2): 876- 879

2 Abdollahi A . Powerful p-groups have non-inner automorphisms of order p and some cohomology. J Algebra, 2010, 323 (3): 779- 789

3 Abdollahi A , Ghoraishi M , Wilkens B . Finite p-groups of class 3 have noninner automorphisms of order p. Beitr Algebra Geom, 2013, 54 (1): 363- 381

4 Alperin J L . On a special class of regular p-groups. Trans Amer Math Soc, 1963, 106: 77- 99

5 Alperin J L . Large Abelian subgroups of p-groups. Trans Amer Math Soc, 1965, 117: 10- 20

6 Aschbacher M . Finite Group Theory. Cambridge Studies in Advanced Mathematics, Vol. 10, Cambridge: Cambridge University Press, 1986

7 Benmoussa M T , Guerboussa, Y . Some properties of semi-abelian p-groups. Bull Aust Math Soc, 2015, 91 (1): 86- 91

8 Berkovich Y . A certain nonregular p-group. Sibirsk Mat Ž, 1971, 12: 907- 911 (in Russian)

9 Berkovich Y . On the number of subgroups of given order in a finite p-group of exponent p. Proc Amer Math Soc, 1990, 109 (4): 875- 879

10 Berkovich Y . Groups of Prime Power Order, Vol. 1. De Gruyter Expositions in Mathematics, Vol. 46, Berlin: Walter de Gruyter, 2008

11 Berkovich Y , Janko Z . Groups of Prime Power Order, Vol. 2. De Gruyter Expositions in Mathematics, Vol. 47, Berlin: Walter de Gruyter, 2008

12 Berkovich Y , Janko Z . Groups of Prime Power Order, Vol. 3. De Gruyter Expositions in Mathematics, Vol. 56, Berlin: Walter de Gruyter, 2011

13 Berkovich Y , Janko Z . Groups of Prime Power Order, Vol. 4. De Gruyter Expositions in Mathematics, Vol. 61, Berlin: Walter de Gruyter, 2016

14 Berkovich Y , Janko Z . Groups of Prime Power Order, Vol. 5. De Gruyter Expositions in Mathematics, Vol. 62, Berlin: Walter de Gruyter, 2016

15 Berkovich Y , Janko, Z ., Groups of Prime Power Order, Vol. 6. De Gruyter Expositions in Mathematics, Vol. 65, Berlin: Walter de Gruyter, 2018

16 Bodnarchuk L Yu , Pilyavs’ka O S . On the existence of a noninner automorphism of order p for p-groups. Ukrainian Math J, 2001, 53 (11): 1771- 1783

17 Bray J N , Wilson R A . On the orders of automorphism groups of finite groups, II. J Group Theory, 2006, 9 (4): 537- 545

18 Cartwright M . Bounded conjugacy conditions. Irish Math Soc Newslett, 1984, (12): 14- 21

19 Cartwright M . Class and breadth of a finite p-group. Bull Lond Math Soc, 1987, 19 (5): 425- 430

20 Cutolo G , Smith H , Wiegold J . The nilpotency class of p-groups in which subgroups have few conjugates. J Algebra, 2006, 300 (1): 160- 170

21 Deaconescu M , Silberberg G . Noninner automorphisms of order p of finite p-groups. J Algebra, 2002, 250 (1): 283- 287

22 Du Sautoy M P F , Vaughan-Lee M . Non-PORC behaviour of a class of descendant p-groups. J Algebra, 2012, 361: 287- 312

23 Dyubyuk P E . On the number of subgroups of an Abelian p-group. Izvestiya Akad Nauk SSSR Ser Mat, 1948, 12: 351- 378

24 Dyubyuk P E . On the number of subgroups of certain categories of finite p-groups. Mat Sbornik N.S., 1952, 30 (72): 575- 580 (in Russian)

25 Eick B , Newman M F , O’Brien E A . The class-breadth conjecture revisited. J Algebra, 2006, 300 (1): 384- 393

26 Evseev A . Higman’s PORC conjecture for a family of groups. Bull Lond Math Soc, 2008, 40 (3): 415- 431

27 Faudree R . A note on the automorphism group of a p-group. Proc Amer Math Soc, 1968, 19: 1379- 1382

28 Felsch V . The computation of a counterexample to the class-breadth conjecture for p-groups. In: The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., Vol. 37, Providence, RI: AMS, 1980, 503- 506

29 Felsch W , Neubüser J , Plesken W . Space groups and groups of prime-power order, IV, Counterexamples to the class-breadth conjecture. J London Math Soc (2), 1981, 24 (1): 113- 122

30 Fouladi S , Orfi R . Noninner automorphisms of order p in finite p-groups of coclass 2, when p > 2. Bull Aust Math Soc, 2014, 90 (2): 232- 236

31 Gallian J A . On the breadth of a finite p-group. Math Z, 1972, 126: 224- 226

32 Gaschütz W . Nichtabelsche p-gruppen besitzen äussere p-automorphismen. J Algebra, 1966, 1- 2 (in German)

33 Gavioli N , Mann A , Monti V , Previtali A , Scoppola C M . Groups of prime power order with many conjugacy classes. J Algebra, 1998, 202 (1): 129- 141

34 Glauberman G . Large abelian subgroups of finite p-groups. J Algebra, 1997, 196: 301- 338

35 Glauberman G . Abelian subgroups of small index in finite p-groups. J Group Theory, 2005, 8 (5): 539- 560

36 Glauberman G . Existence of normal subgroups in finite p-groups. J Algebra, 2008, 319 (2): 800- 805

37 Glauberman G . A partial analogue of Borel’s fixed point theorem for finite p-groups. J Algebra, 2016, 450: 398- 457

38 Glauberman G , Mazza N . p-groups with maximal elementary abelian subgroups of rank 2. J Algebra, 2010, 323 (6): 1729- 1737

39 González-Sánchez J , Jaikin-Zapirain A . Finite p-groups with small automorphism group. Forum Math Sigma, 2015, 3: e7, 11 pp.

40 Green D , Héthelyi L , Lilienthal M . On Oliver’s p-group conjecture. Algebra Number Theory, 2008, 8 (2): 969- 977

41 Green D , Héthelyi L , Mazza N . On Oliver’s p-group Conjecture, II. Math Ann, 2010, 347 (1): 111- 122

42 Green D , Héthelyi L , Mazza N . On a strong form of Oliver’s p-group conjecture. J Algebra, 2011, 342: 1- 15

43 Green D , Lynd J . Weak closure and Oliver’s p-group conjecture. Israel J Math, 2013, 197 (1): 497- 507

44 Higman G . Enumerating p-groups, I, Inequalities. Proc London Math Soc 1960, 10: 24- 30

45 Higman G . Enumerating p-groups, II, Problems whose solution is PORC. Proc London Math Soc (3), 1960, 10: 566- 582

46 Hua, L.-K. , Some “Anzahl” theorems for groups of prime power orders. Sci Rep Nat Tsing Hua Univ., 1947, 4: 313- 327

47 Hughes D R . Partial difference sets. Amer J Math., 1956, 78: 650- 674

48 Hughes D R . A problem in group theory, Research Problems, No. 3. Bull Amer Math Soc, 1957, 63: 209

49 Jamali A R , Viseh M . On the existence of noninner automorphisms of order two in finite 2-groups. Bull Aust Math Soc, 2013, 87 (2): 278- 287

50 Janko Z . Finite nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4. J Algebra, 2009, 321 (10): 2890- 2897

51 Janko Z . Finite p-groups with many minimal nonabelian subgroups. J Algebra, 2012, 357: 263- 270

52 Janko Z . Finite p-groups of exponent p^e all of whose cyclic subgroups of order p^e are normal. J Algebra, 2014, 416: 274- 286

53 Janko Z . Finite p-groups with some isolated subgroups. J Algebra, 2016, 465: 41- 61

54 Khukhro E I . On a question of G. Glauberman about a replacement theorem for finite p-groups J Algebra, 2001, 241 (1): 247- 258

55 Knoche H -G . Über den Frobenius’schen Klassenbegriff in nilpotenten Gruppen. Math Z, 1951, 55: 71- 83 (in German)

56 Knoche H.-G . Über den Frobeniusschen Klassenbegriff in nilpotenten Gruppen, II. Math Z, 1953, 59: 8- 16 (in German)

57 Kulakoff A . Über die Anzahl der eigentlichen untergruppen und der elemente von gegebener ordnung in p-gruppen. Math Ann, 1931, 104 (1): 778- 793 (in German)

58 Lee S . A class of descendant p-groups of order p⁹ and Higman’s PORC conjecture. J Algebra, 2016, 468: 440- 447

59 Leedham-Green C R , McKay S . The Structure of Groups of Prime Power Order. London Mathematical Society Monographs, New Series, Book 27, Oxford: Oxford University Press, 2002

60 Leedham-Green C R , Newman M F . Space groups and groups of prime-power order, I. Arch Math (Basel), 1980, 35 (3): 193- 202

61 Leedham-Green C R , Neumann P M , Wiegold J . The breadth and the class of a finite p-group. J London Math Soc (2), 1969, 1: 409- 420

62 Liao S T , Liu W J . A introduction to the theory of homotopy, Beijing: Peking University Press, 1980

63 Liebeck H . Outer automorphisms in nilpotent p-groups of class 2. J London Math Soc, 1965, 40: 268- 275

64 Longobardi P , Maj M . On p-groups of breadth two. Algebra Colloq, 1999, 6 (2): 121- 124

65 Longobardi P , Maj M , Mann A . Minimal classes and maximal class in p-groups. Israel J Math, 1999, 110: 93- 102

66 Lynd J . 2-subnormal quadratic offenders and Oliver’s p-group conjecture. Proc Edinb Math Soc (2), 2013, 56 (1): 211- 222

67 Macdonald I D . The breadth of finite p-groups, I. Proc Roy Soc Edinburgh Sect A, 1977/78, 78 (1/2): 31- 39

68 Macdonald I D . Groups of breadth four have class five. Glasgow Math J, 1978, 19 (2): 141- 148

69 Macdonald I D . Some p-groups of Frobenius and extra-special type. Israel J Math, 1981, 40 (3/4): 350- 364

70 Mann A . Some questions about p-groups. J. Austral Math Soc Ser A, 1999, 67 (3): 356- 379

71 Mann A . Elements of minimal breadth in finite p-groups and Lie algebras. J Aust Math Soc, 2006, 81 (2): 209- 214

72 Mann A . The derived length of p-groups. J Algebra, 2000, 224 (2): 263- 267

73 Mann A . Spreads and nilpotence class in nilpotent groups and Lie algebras. J. Algebra, 2015, 421: 12- 15

74 Martino J , Priddy S . Unstable homotopy classification of BG∧ p. Math Proc Cambridge Philos Soc, 1996, 119 (1): 119- 137

75 Mazurov V D , Khukhro E I (eds.). Unsolved Problems in Group Theory, The Kourovka Notebook, No. 16. Novosibirsk: Russian Academy of Sciences Siberian Division, Institute of Mathematics, 2006

76 Meierfrankenfeld U , Stellmacher B , Stroth, G . Finite groups of local characteristic p: an overview. In: Groups, Combinatorics & Geometry (Durham, 2001), River Edge, NJ: World Sci. Publ., 2003, 155- 192

77 Neumann M F . On coclass and trivial Schur multiplicator. J Algebra, 2009, 322 (3): 910- 913

78 O’Brien E A . The p-group generation algorithm. J Symbolic Comput, 1990, 9 (5/6): 677- 698

79 Oliver B . Equivalences of classifying spaces completed at odd primes. Math Proc Cambridge Philos Soc, 2004, 137 (2): 321- 347

80 Oliver B . Equivalences of Classifying Spaces Completed at the Prime Two. . Mem Amer Math Soc, No. 848, Providence, RI: AMS, 2006

81 Parmeggiani G , Stellmacher B . p-groups of small breadth. J Algebra, 1999, 213 (1): 52- 68

82 Qu H P , Sun Y , Zhang Q H . Finite p-groups in which the number of subgroups of possible order is less than or equal to p3. Chin Ann Math Ser B, 2010, 31 (4): 497- 506

83 Ruscitti M , Legarreta L , Yadav M K . Non-inner automorphisms of order p in finite p-groups of coclass 3. Monatsh Math., 2017, 183 (4): 679- 697

84 Schenkman E . The existence of outer automorphisms of some nilpotent groups of class 2. Proc Amer Math Soc, 1955, 6: 6- 11

85 Schmid P . Normal p-subgroups in the group of outer automorphisms of a finite p-group. Math Z, 1976, 147 (3): 271- 277

86 Schmid P . A cohomological property of regular p-groups. Math Z, 1980, 175 (1): 1- 3

87 Shabani-Attar M . On a conjecture about automorphisms of finite p-groups. Arch Math (Basel), 2009, 93 (5): 399- 403

88 Shalev A . The structure of finite p-groups: effective proof of coclass conjectures. Invent Math, 1994, 115 (2): 315- 345

89 The GAP Group, GAP—Groups, Algorithms, Programming—A System for Computational Discrete Algebra, Version 4.4.10, 2007

90 Tong W T . A introduction to homological algebra, Beijing: Higher Education Press, 1998

91 Tuan H F . An Anzahl theorem of Kulakoff’s type for p-groups. Sci Rep Nat Tsing Hua Univ Ser A, 1948, 5: 182- 189

92 Vaughan-Lee M R . Breadth and commutator subgroups of p-groups. J Algebra, 1974, 32: 278- 285

93 Vaughan-Lee M R . Groups of order p⁸ and exponent p. Int J Group Theory, 2015, 4 (4): 25- 42

94 Vaughan-Lee M R . Non-PORC behaviour in groups of order p7. J Algebra, 2018, 500: 30- 45

95 Wiegold J . Groups with boundedly finite classes of conjugate elements. Proc Roy Soc London Ser A, 1957, 238: 389- 401

96 Wilkens B . 2-groups of breadth 3. J Algebra, 2007, 318 (1): 202- 224

97 Xu M Y . The power structure of finite p-groups. Bull Austral Math Soc, 1987, 36 (1): 1- 10

98 Xu M Y . Some problems on finite p-groups. Adv Math (China), 1985, 14 (3): 205- 226 (in Chinese)

99 Xu M Y , Qu H P . Finite p-groups. Beijing: Beijing University Press, 2010

100 Xu X Z . A note on Oliver’s p-group conjecture. J Algebra, 2018, 507: 421- 427

101 Zhang J P . Finite groups and fusion systems. Sci Sin Math, 2016, 46 (6): 769- 780 (in Chinese)

102 Zhang Q H , An L J . The structure of finite p-groups, Vol. 1. Beijing: Science Press, 2017

103 Zhang Q H , An L J . The structure of finite p-groups, Vol. 2. Beijing: Science Press, 2017

104 Zhang Q H , Qu H P . On Hua-Tuan’s conjecture. Sci China Math, 2009, 52 (2): 389- 393

105 Zhang Q H , Qu H P . On Hua-Tuan’s conjecture II. Sci China Math, 2011, 54 (1): 65- 74