Permutation Groups: Misprints, Corrections, Improvements

Corrections:

• Page 7, line 13: Not really a mistake but misleading: in place of "congruent to m mod p^a", read "equal to m".
• Page 17, start of 1.13: The number of generators is not necessarily equal to the number of points! The generators should be g₁, …, g_r.
• p.21, l.9: "Turrull" should be "Turull". (Spotted by P. P. Pálfy)
• Page 30, Exercise 1.19(b): The congruence q≡1 (mod 4) should be replaced by q≡−1 (mod 4). Also, the assumptions of the exercise are stronger than needed. I am grateful to Dávid Szabó for this, and with his permission I have posted his amendment and solution here.
• Page 30, Exercise 1.21: the question should say "of degree greater than 5". Note that it applies to finite and infinite permutation groups. (Spotted by Alice Devillers.)
• Page 32, Exercise 1.30(b): "… fixed point set of K" (not G). (Spotted by Pablo Spiga.)
• Page 34, Exercise 1.36: for the "if" part of the question, you must assume that G is a transitive permutation group – this follows from the sharp transitivity of S when you are going in the "only if" direction. (Spotted by Pablo Spiga.)
• Page 50, line 12: this is not very clear. The map g → g^k/d is a bijection from the dth powers to the kth powers in G, and every kth power is a dth power, so the two sets are the same. (Spotted by Pablo Spiga.)
• Page 50, line −7: χ(g) should be χ(g^k).
• Pages 54,55: The inner product of π^(n−2,1,1) with itself should be 7, not 6 (in two places). The conclusion of the argument is correct.
• Page 63, line −9: should say "subset of Ω²". (Spotted by Robin Whitty.)
• Page 76, line 2: g(θ)=1, g(φ)=0 (not f).
• Page 77, line 2: +(k−mu) should be −(k−mu). (Spotted by P. P. Pálfy)
• Page 83, line 4: PΣL(3,5²) should be PΣU(3,5²). (Spotted by P. P. Pálfy)
• Page 101, 2nd line of proof of 4.3: G should be N. (Spotted by Nick Cavenagh.)
• Page 101: a more elementary argument to finish the proof of Theorem 4.3 is given here. (Spotted by Ram Abhyankar.)
• The reference to the classification of the affine 2-transitive groups (p.110) is inadequate and should be supplemented with the following papers:
  • Cristoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II. J. Algebra 93 (1985), 151–164.
  • Martin W. Liebeck, The affine permutation groups of rank three, Proc. London Math. Soc. (3) 54 (1987), 477–516.
• Pages 135-138, Section 5.3: In the discussion of first-order logic, I neglected to say that the binary relation = is assumed to be among the logical symbols, and we always assume that its interpretation is the usual one of identity.
• Page 139, Section 5.5, line 5: delete "countable". (Spotted by P. P. Pálfy)
• Page 164, Exercise 5.23(b): kⁿ should be k^n-1. (Spotted by Nathan W. Lemons)
• Page 166, line 14: it should read "is reflexive and transitive". (Spotted by Pablo Spiga)
• Page 166, last line: "cofinitary" should be "finitary". (Spotted by P. P. Pálfy)
• Page 170, line 6: For "sharply k-transitive" read "sharply k-set-transitive".
• Page 180, line 21: B a maximal block meeting Δ. (Spotted by P. P. Pálfy)
• Page 188, line 9: delete the words "of the same order".
• Page 194, line 3: delete p.
• Page 198, Exercise 7.4: should say "Table 7.4" (not "Table 6.4"). (Spotted by Alberto Basile.)
• Page 200, Reference 15: Aron Bereczky's paper is in the Bulletin of the London Mathematical Society, not the Journal. (Spotted by Pablo Spiga.)

Updated references:

[9] László Babai and Peter J. Cameron, Automorphisms and enumeration of switching classes of tournaments, Electronic J. Combinatorics 7 (2000), #38 (25pp.)
[12] R. A. Bailey, Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 2004.
[48] Peter J. Cameron and Csaba Szabó, Independence algebras, J. London Math. Soc. (2) 61 (2000), 321-334.
[81] L. A. Goldberg and M. R. Jerrum, The "Burnside process' converges slowly, in: Proceedings of Random 1998, Randomisation and Approximation Techniques in Computer Science, Lecture Notes in Computer Science 1518, Springer-Verlag, pp. 331-345.
[119] Martin W. Liebeck and Aner Shalev, Simple groups, permutation groups, and probability, J. Amer. Math. Soc. 12 (1999), 497-520.
[131] Dugald Macpherson, Sharply multiply homogeneous permutation groups, and rational scale types, Forum Math. 8 (1996), 501-507.
[165] Ákos Seress, Permutation Group Algorithms, Cambridge Tracts in Mathematics 152, Cambridge University Press, 2003.

Please email corrections to me: p.j.cameron(at)qmul.ac.uk or (preferred) pjc20(at)st-andrews.ac.uk

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Peter J. Cameron
17 January 2016.