permutation groups with multiply-transitive suborbits, ii - CiteSeerX

PERMUTATION GROUPS WITH MULTIPLY-TRANSITIVE SUBORBITS, II PETER J. CAMERON In a previous paper with the same title [1], I considered the following situation; G is a primitive, not doubly transitive permutation group on Q, in which the stabiliser Ga of a point a acts doubly transitively on an orbit F(a), where |F(a)| = v. Manning [3] showed that, if v > 2, then Ga has an orbit larger than F(a). Indeed, with A = F* o T (see [1] for notation), it is easy to see that A(a) is a Ga-orbit and |A(a)| = v(v-l)/k, with k < v-l if v > 2. In [1], I showed that k < \{v-\) if v > 5. Furthermore, if Ga is triply transitive on F(a), then k = 0(i>*); if Ga is quadruply transitive on F(a), then k < 2. I remarked there that the truth is probably stronger that these results suggest, since essentially only two situations are known in which k > 2; these are the Mathieu group M 22 (or its automorphism group) with v = 16, k = 4, |Q| = 77, and the Higman-Sims group HS (or its automorphism group) with v = 22, k = 6, |ft| = 100. The main result of this paper is a substantial improvement on Theorem 2.2 of [1]: in the above situation, k = O(v*). This is proved by using more efficiently the information in [1], together with a lemma about the number of blocks meeting a given block in a block design. Further, I show that if either G has rank 3 on Q or Ga has rank 3 on A (a), then k = O(v*). In the latter case, a more precise result is obtained, identical with the conclusion of Theorem 3.1 of [1]. 1. A theorem on block designs A block design consists of a set of v points and a set of b blocks, with a relation called incidence between points and blocks, having the properties that any block is incident with k points and any two points with A blocks, where X > 0 and k < v— 1. The number of blocks incident with a point is also a constant, denoted by r. Often a block is identified with the set of points incident with it. I. In a block design with parameters v, k, X, r, b (as above), the number of blocks having non-empty intersection with a given block is at least k(r—l)2/ (kX—k—X+r). If a block B has non-empty intersection with exactly k(r—l)2/ (kX-k-X+r) other blocks, then \BnB'\ = 0 or (kX-k-X + r)l(r-\) for any block B' *B. THEOREM

Proof. Let B be a block, and xlt ..., xd the sizes of the non-empty intersections of B with other blocks. In what follows, all summations are over the range 1 < i < d. Received 19 July, 1973. [BULL. LONDON MATH. SOC, 6 (1974), 136-140]

PERMUTATION GROUPS WITH MULTIPLY-TRANSITIVE SUBORBITS, II

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Count in two ways the number of choices of j points of B and a block B' # B which intersects B non-trivially and contains the chosen points, for j = 0, 1, 2:

£*,(*,-l) = *(fc-l)(A-l). So, if x denotes a real variable, JXx-x,)2 = dx2-2k(r-l)x Clearly the left-hand side is non-negative for all x; so the quadratic form on the right is positive semi-definite, and hence k2{r-\)2 ^dk{kk-k-k + r). If equality holds, then A: = (kk-k — k + r)/(r— 1) is a solution of the equation dx2-2k(r-l)x + k(kk-k-k + r) = Q; so, with this value of x,

Ux-xd2 = 0. This implies xt = x for i = 1, ..., d. I mention two well-known special cases of Theorem 1. It clearly implies k(r-l)2/(kk-k-k + r) ^ b- 1. Using vr = bk and (v- 1) k = r(k — 1), this becomes

so r ^ k, and equivalently b ^ v {Fisher's inequality). Furthermore, if equality holds then any two blocks have A common points. (Such a design is called symmetric.) A parallelism (or resolution) in a block design is a partition of the block set into " parallel classes " each of which partitions the point set. Since each parallel class contains b/r blocks and parallel blocks are disjoint, a design with parallelism satisfies k(r-l)2/(kk-k-A+r) ^ b-b/r. Again using vr = bk and r{k—\) = (v— 1)A, this becomes

(r-k-A){b-r){r-l)>0, so r > k + k, and equivalently b ^ v + r—l (Bose's inequality). If equality holds, then any two non-parallel blocks have kA/(k+X-1) common points. (Such a design is called affine.) Indeed, Theorem 1 generalises a result of Mavron [4] on affine designs.

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2. The main theorem THEOREM 2. / / G is a primitive but not doubly transitive permutation group on Cl, Ga is doubly transitive on F(a), |F(a)| = v, F* o F = A, and |A(a)| = v(v- \)/k, then

v(v-\)k(k2-3k+v)2

^

2{v-k)4(v-k-\)2,

which implies k < (2v)*. Proof. Clearly the theorem is true for k = 1; so assume k > 1. Recall from [1] that Ga is doubly transitive on F*(a), the suborbit paired with F(a). As in [1], a block design is constructed as follows: the point set is F*(a), the block set is A(a), and the point y and block 5 are incident if and only if (y, 5) e T. The parameters v and k were chosen to agree with the block design parameters denoted by the same By Theorem 1, the letters; in addition, X = k-\, r = v-\, and b = v(v-\)/k. number of blocks disjoint from a given block is at most v(v-l)/k-\~k(v-2)2/(