COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

174 Charles C. Sims
the intersection being taken over all g in G such that {cc, /?} E d*. Then
Y’* = ? (A*Y
and y* is a set of imprimitivity of G*. By Lemma 3.2 1~1 = 2. Thus there is
a g in G such that {a, Is} C Ag and y $ AZ. Then U = g-‘Vg s Gti and
u-4 1.
The proof of Theorem 3.1 is completed as in the proof of Theorem 18.2
of [16].
We wish now to consider the consequences of the assumption that G*
is imprimitive. If we are assuming that G cannot be represented nontrivially
on fewer than rz = 1521 points, then G* cannot have a set of imprimitivity
with more than (n- 1)/2 elements. Let B = {Ei 11 s i 4 b} be
a non-trivial system of imprimitivity for G*. Let Ai denote the union
of the elements in Et and let @i be the undirected graph (Ai, Ei). (Q, B) is
a block design, where a point a is incident to Et if and only if a E Ai.
(It is possible to have Ai = Ai with i + j.) Let o, b, k, r, il be the parameters
of this block design. The graphs @i are all isomorphic and the
automorphism group of any one of them is transitive on vertices and edges.
Thus there is a positive integer d such that each vertex of @i is connected
to exactly d other vertices. The integers V, b, k, r, I, d will be called the
parameters of the system of imprimitivity B.
THEOREM 3.4. The parameters of a system of imprimitivity B for G* satisfy
the following conditions:
(1) bk = ru,
(2) r(k-1) = 3(v-l),
(3) dk is even and bdk = w(w- l),
(4) rd = w-l,
(5) dA = k-l.
Proof. Let B = {Ei 1 1 4 i e b} and let @i be as defined above. (1) and
(2) follow from the fact that (G, B) is a block design. Since pi has kvertices
and each vertex is connected to d other vertices, I&/ = dk/2. Therefore
or
1 Q* / = V(V- 1)/2 = b 1 Et 1 = bdk/2,
bdk = v(v- 1).
(4) follows from (1) and (3). Finally (2) and (4) imply (5).
We now prove an analogue of Theorems 17.4 and 17.5 of [16].
THEOREM 3.5. Let G be doubly transitive on Q, let a and B be distinct points
of Q, and let the orbits of Gas be {a} = Al, {/I} = AZ, As, . . ., A,. Set
nt = [Ai/ and suppose 1 = nl = n2 =s n3 < . . . == n,. Then at least one of
the following holds:
Computational methods for permutation groups 175
(1) if 3 =Z i e t, then nt < n3ni- 1 and (nt, n,) =+ 1,
(2) there exists a non-trivial system of imprimitivity for G* for which
d=- 1,
(3) G is sharply doubly transitive.
Proof. If n, = 1, then G is sharply doubly transitive. If n, =- 1 and n3 = 1,
then by Theorem 3.1 G is an automorphism group of a non-trivial block
design with J = 1. If ~1, . . ., ?+$, are the blocks of this design, then
bf&
. . . , y$} is a non-trivial system of imprimitivity for G* for which
d = 1 vi I- 1 * 2. Thus we may assume Iti z= 2 for i z= 3. Suppose there exists
an i * 3 such that (ni, n,) = 1. For any two distinct points y and 6 of 52 let
r(y, S) denote the union of those orbits of G,,6 with length n,. Clearly
Qy, 4 = Jl4 Y). No wchoose y E Ai. From the proof of Theorem 17.5
of [16] applied to G, we see that r(a, ,!?) = I’(a, r). Thus we can obtain
a non-trivial equivalence relation N on Q* by defining {y, S} TV {a, q}
if and only if r(y, S) = J’(E, r]). The equivalence classes of cv form a
non-trivial system of imprimitivity for G* for which d z= ni z= 2. If there is
an i Z= 3 such that ni =- nsni-1, we define r(y, 6) to be the union of all
orbits of GYd with length at least Iti. If y E AS, then by the proof of Theorem
17.4 of [16] applied to G, r(a, ,@ = r(a, 7) and we can proceed as before.
THEOREM 3.6. Let v, b, k, r, I, d be the parameters of a non-trivial system
of imprimitivity of G*. Any one of the following implies that G is sharply
doubly transitive or an automorphism group of a non-trivial block design
for which il = 1:
(1) d = 2,
(2)d= landk=v-1,
(3) d = 1, k < 6, and v is odd.
Proof. We shall show that if any one of the conditions (I), (2), or (3)
holds, then, for any two distinct points a and b of 0, GDls fixes at least 3
points. It will follow by Theorem 3.1 that G is sharply doubly transitive
or an automorphism group of a non-trivial block design with il = 1. Let
B = {Ei 1 1 =z i == b} be the system of imprimitivity for G* and let the
graphs pi be defined as before. Suppose first that d = 2. Given a + /?
there exists a unique y + p such that {a, ,9} and {a, y} are edges of the
same &. Therefore GaB fixes y. Suppose now that d = 1 and k = v- 1.
Then b = u and for each a E Sz there is a unique @i such that a is not a
vertex of &. Given p + a, GDP must fix the unique point y such that {p, y}
is an edge of pi. Finally, suppose d = 1 and k s 6. Then k = 4 or 6.
If k = 4, let Er = {{a, /?}, {y, S}}. Gzs maps {y, S} into itself and if G
is not an automorphism group of a block design for which I = 1, then
Gti is a 2-group. If in addition v is odd, then GNP must fix a third point.
A similar argument takes care of the case k = 6.
THEOREM 3.7. Let G be doubly transitive on Q, let a and p be distinct
points of Q, and let H be the subgroup of G mapping {a, /?} into itself. If G,