A CLASS OF DOUBLY T R A N S I T I V E
PERMUTATION GROUPS
By MICHIO SUZUKI
I will consider here a class of doubly transitive permutation groups which
have been the object of recent investigations. The problem of determining
all multiple transitive permutation groups is an old one and it still awaits a
solution. I do not consider this problem at all; rather I consider a particular
class of groups.
It has been known that certain classes of permutation groups are characterized by their group theoretical properties. I will mention a few results
along this line. I use the following notation. A permutation group on a set
O is usually denoted by G. I consider only a transitive permutation group
on a finite set. If a,b,... are distinct elements of O, I denote by Ga, &,... the
subgroup of G consisting of permutations leaving each of these elements
a,b,....
A Frobenius group is a transitive permutation group satisfying Ga, 0, = {1}
for any pair a,b of elements of Q,. The class of Frobenius groups is characterized by several group theoretical properties. If H is the subgroup Ga of a
Frobenius group G, H satisfies the property
Hdg-1Hg
= {l}
for aU
g£G-H.
This property characterizes the class of Frobenius groups: namely a group G
containing a subgroup H with the above property is a Frobenius group. A
famous theorem of Frobenius asserts that in a Frobenius group G the totality
of elements which moves every element of O forms together with the identity
a normal subgroup N which is called the Frobenius kernel of G and satisfies
the property that
CG(x)^N
for
x*l,xeN.
Here CG(x) denotes the eentralizer of x in G. Again this is a characteristic
property of the class of Frobenius groups. The structure of Frobenius groups
is fairly well known. The factor group GjN by the Frobenius kernel has been
studied by Zassenhaus who classified the possible structure of GjN, and the
structure of N itself is nilpotent by a theorem of Thompson.
Zassenhaus has studied a class of triply transitive groups G satisfying
Ga, &.c = {l}- This is a class of triply transitive groups of smallest possible
order, and the result of Zassenhaus is that groups in this class are the
groups of all the linear fractional transformations of one variable over some
near fields, whose structures are completely determined.
I considered a little wider class of groups which are called (Z)-groups. A
(Z)-group is a doubly transitive permutation group G satisfying the following
two conditions:
(l)G 0 , 6 . c = {l}and
(2) There is no regular normal subgroup.
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M. SUZUKI
The second restriction is not essential. In fact if a doubly transitive permutation group satisfies (1) but not (2) then the group is a group of semilinear transformations
x'=axa+b
over a finite near field. The structure of (Z)-groups is studied by Feit,
Ito and myself. It is convenient to denote the degree of G by 1 +g. Then the
group Ga is a Frobenius group on q elements. Hence Ga contains a normal
.subgroup Q of order exactly q. By using a theory of characters Feit proved
that for a (Z)-group q must be a power of a prime number. The case when
this prime number p is greater than 2 is settled by Ito, again by using a
theory of characters. The result in this case is that a (Z)-group G on a set Q
is a subgroup of index < 2 of a triply transitive group on Q which satisfies
the condition ( 1 ). Hence again G is a group of linear fractional transformations
over some near fields. The case whenjp=2 caused a difficulty. I call a (Z)group with p=2 a (ZT)-group. It is known that the totality of linear fractional transformations over a field of characteristic 2 is a (ZT)-group, when
it is considered as a permutation group on points of the projective line over
the field. Recently I discovered a class of new simple groups. They are
defined as a subgroup of GL (4, q) where g is a power of 2 and is non square.
Originally they are given as a group generated by a certain set of 4 x 4
matrices. Perhaps it is easy to define them as a set of fixed elements by a
special automorphism of the classical group Sp(2), which in a sense exchanges two dots of Dynkin Diagram of B2. This new definition is due to
Ree and Steinberg. One property of this class of groups is that every member
is a (ZTJ-group. Tits remarked that in the three dimensional projective
space over GF(q) there is a certain oval consisting of 1 +q2 points and the
group in question is defined as the totality of projective transformations
leaving this oval invariant. The permutation representation on points of
this oval is doubly transitive and satisfies the properties (1) and (2) if q > 2.
I t was rather difficult to prove that there are no other (ZT)-groups.
A class of (ZT)-groups is characterized by some group theoretical properties. Namely one has the following theorem. A group G is a (Z!F)-group if
and only if G is a non-abelian simple group containing a subgroup H such
that
(i) the center of H has even order, and
(ii) C%(x)^H lot
x^l,x£H.
Here C%(x) denotes the generalized centralizer: i.e.
C*G(x) = {y\y~1xy=x
or x-1}.
The conditions (i) and (ii) can be stated in weaker forms. For example (ii)
is required only for involutions in H and for other elements the conditions
G%(x)^NG(H)
are sufficient.
A (ZT)-group satisfies other interesting group theoretical properties. In
a (ZT)-group G the centralizer of every non-identity element is nilpotent.
DOUBLY TRANSITIVE PERMUTATION GROUPS
287
This property is of course shared by other groups. However it can be proved
that if a non-abelian simple group G satisfies the above property, then G is
either a (ZjP)-group or one of the linear groups.
The simple groups in the first class discovered by Ree also admit a doubly
transitive representation. Ree's groups do not satisfy 6r a i ö > c ={l} for some
triple a,b,c but the group Ga%btC is very small in any case. Both Ree's
groups and (Z)-groups have the following property in common:
(*) The group Ga contains a normal subgroup Q, which is transitive and
regular on O — {a}.
I studied a class of doubly transitive permutation group G of degree n
satisfying the condition (*). The main result is the following theorem.
Let G be as above. If moreover the degree n is odd and if the group GajQ is a
solvable group of odd order, then G is isomorphic to one of the following three
groups: (i) an extension of a (ZT)-group, (ii) an extension of a unitary group of
dimension 3 over a field of characteristic 2 or (iii) an extension of the group of
all linear transformations over a near field.
Ree's groups do not come in because I assumed that the degree is odd.
In all cases extensions are obtained by applying automorphisms of the
underlying (near) field.
The proof of the above theorem requires many tools, including the theory
of characters and of transfers. A recent theorem of G. Higman on the structure of a particular class of 2-groups is used to determine the structure of
Sylow 2-groups of G. The ideas in proof are quite similar to the ones used to
determine the structure of (ZT)-groups. The last case (iii) is a sort of a
degenerated case. One has the case (iii) if and only if the group Q contains
only one involution.
The assumption (*) seems quite essential in this study. Also the assumption
of solvability of Ga/Q is used to determine the structure of Ga§ b. Although
my method does not apply to the groups of even degree, there are several
indications that one might be able to classify all groups with the condition
(*)•
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