(2001.11.23)

Chapter 1

11:-10  read ``on each of its orbits of length > 1,''

13:13-15  read ``Suppose that G is a group acting primitively on a set W and that D is a proper subset of W containing at least two points.  Show that for each pair of distinct points ...''

13:18  add ``[Hint: Show that the relation a ~ bÛ(for all x Î G, { a,b}ÇD^x = { a,b} or Æ) is a G-congruence.]''

17:-3  read ``If D,D^¢ Î S are fixed setwise by H, then ''

19:21  read  ``If fix(G_a) is finite, show it is a block for G.''

23:9  read ``and let a Î W. ''

27:-10  read ``acting transitively on a set W''

Chapter 2

35:20  read ``Hence show that S_n acts ...''

48:-5  read ``S_p^m ( @ Sym(D^m))  ''

53:15  read ``and G_¥0 transitive on the nonzero elements of F.''

57:-18  read ``the set of projective points ... "

63:-16  read ``PGL₂(5) @ S₅''

Chapter 3

66:-9  read  ``the diagonal orbit D₁: = { (a,a)  | a Î W} ; the other orbitals are called nondiagonal.''

68:9  read ``H: = á t ñ ''

70:-16  delete ``that G is finite,''

75:-19 read ``A₃ is a composition factor''

84:-1 and 85:1,2  replace ``2-cycle'' by ``3-cycle'' and ``p ¹ 2'' by ``p ¹ 3''

93:-8  read ``(a₁+...+a_k)^p = a₁^p+...+a_k^p ''

102:20  read ``w Î W''

Chapter 4

110:4  read ``each point stabilizer of H is its own normalizer in H, ''

124:-4  read ``H is a transitive normal subgroup''

Chapter 5

163:4  read  ``5.5.2  Using the fact that l(s+1) ³ (2s-4)/3 ...''

170:1-4 read ``5.7.3  Show that A₆ is isomorphic to SL₂(9) modulo its centre.  Hence l(6) = 2.''

170:5-6  read ``5.7.4  Show that there is no field F for which SL₂(F) contains a finite preimage G of A₇.  (However, A₇ is isomorphic to a section of SL₃(25), and so l(7) = 3.)''

170:13  read ``For all k ³ 5, l(k) ³ (2k-6)/3.''

172:8  read ``...  Since k ³ 8, we have d ³ 3''

172:21-22  read ``... shows that d-2 ³ {2(k-3)-6}/3 and hence d ³ (2k-6)/3 as required. ...''

172:after the last line add the following paragraph:

      ``Note that if d = 3 then the Jordan form for x cannot consist of a single block.  Indeed, the centralizer of such a block is a group of upper triangular matrices and hence solvable, but we know that C_G(x) is not solvable.''

173:2  delete ``(since d ³ 4)''

Chapter 7

210:-2  read ``the stabilizers G_a1a2...ak of k points''

217:10  read ``Theorem 7.2C shows''

217:13  read ``finite Frobenius''

Chapter 8

256:15  read ``has order | W| for c = À₀ and order at most | W| ^c for À₀ < c £ |W|.''

262:12  read ``Theorem 3.3C shows ''

263:  replace the second paragraph by:

Let G £ FSym(W) be residually finite. We have to show that every orbit of G is finite.  Suppose the contrary and let S be the union of the infinite G-orbits.  Put K: = G_(W\S).

First note that if H £ G has finite index in G, then S is a union of infinite H-orbits.  Indeed, if g Î S, then |g^H| = | H:H_g| ³ | G:G_g| /| G:H| .

We next show that K must be transitive on each infinite G-orbit G.  Fix a,b Î G with a ¹ b and choose x Î G such that a^x = b;  we must show that a^z = b for some z Î K.  Put D: = supp(x)ÇS and F: = supp(x)\D.  Since each point in the finite set F lies in a finite G-orbit, G_(F) has finite index in G, and so all the G_(F)-orbits in S are infinite.  Thus Theorem 3.3C shows that there exists y Î G_(F) such that the finite subset D Í S satisfies D^yÇD = Æ.  Since the supports of x and y on the invariant subset W\S are disjoint, z: = xyx^-1y^-1 leaves all points in W\S fixed, and so z lies in K.  On the other hand, b^y Î D^y Í S\D and so b^y Ï supp(x).  Therefore a^z = (b^y)^x-1y-1 = b as required.  This proves the transitivity of K on each infinite G-orbit.

Finally, note that for each subgroup H of finite index in K, Lemma 8.3C(i) shows that (K^S)^¢ £ H^S and so K^¢ £ H.  Since K is a subgroup of a residually finite group G, K is also residually finite, and so the intersection of all subgroups of finite index in K must be 1.  Thus K^¢ = 1 and so K is abelian.  However, if G is an infinite K-orbit, then Lemma 8.3C(ii) applied to K^Gshows that Z(K^G) = 1.  Thus K^G = 1 contradicting the transitivity of K on G.  This completes the proof.

Remark 1  This proof is based on P.M. Neumann, ``The  structure of finitary permutation groups'', Archiv Math. 27 (1976) 3-17.

Appendix B  (These corrections are due to Heiko Theissen)

In Table B.2 the ranks of the normalizers of the following groups should be corrected:

A₉ (degree 840): rank 9;  L₂(5²) (degree 325): rank 10;  L₃(2²).3 (degree 960): rank 10;  L₃(2²).2 (degree 336): rank 6;  U₃(2²) (degree 208): rank 4 and (degree 416): rank 5;  S₄(2²).4 (degree 425): rank 5;  Sz(2³) (degree 560): rank 7;  M₁₂ (degree 495): both of rank 8.

Also the normalizer of H = S₄(2³) (degree 585) should be H.3.

In Table B.4 the following counts should be corrected:

Degree 91: there is only one cohort of type C (L₃(9) is incorrectly listed twice)

Degree 244: there is only one cohort of type B

Degree 585: there is only one cohort of type E

• Thanks to Vahid Abdoly, Ulrich Felgner, Richard Lyons, Amtul Khan, Dan Segal and Heiko Theissen for bringing some of these errors to our attention..We should appreciate learning of any other errors.