COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

138 Marshall Hall Jr. Simple groups of order less than one million
PSL,(q) is the projective special linear group of dimension n over GF(q).
Here PSL,(q) is the group of n-dimensional matrices of determinant 1 over
GF(q) modulo its center. PSLz(q) is of order +q(q2- 1) for q odd and of
order q(q2- 1) for q = 2’. PSL,(q) is of order q3(q3- l)(q2- 1)/z where z
is 1 unless there is an element of order 3 in GF(q) in which case z = 3.
Sp2n(q) is the symplectic group of order q”’ fi (q2’- 1).
i=l
U,(q) is the unitary group of order ’T fi(q’-(-l)‘), t = (n,q+l).
i=2
A, is the alternating group of n letters.
The simple Mathieu groups M11, M12, and M22 come within the range
of this search. The Suzuki groups Su(q) with q = 22n+1 == 8 are of order
qz(q2+ l)(q- 1). Here Su(8) of order 29,120 is the only Suzuki group in the
range. The Janko group of order 175,560 is still an isolated group and will
be called merely the Janko group. No simple groups of the Chevalley types
[15] or Rimhak Ree’s [27,28] occur in the range examined.
The known 56 simple groups of order less than one million are: 28 groups
PSL2(p), p a prime, p = 5, 7, . . . , 113. The other 28, listed by their order are:
Group type
PSLz(9) = AB
PSL2(8>
A7
PSLz(l6)
PSLB(3)
U3(3)
PSL2(25)
Ml1
PSLz(27)
PSL4(2) = A8
PSL3(4)
@4(3) = u4(2)
W3)
P&(32)
PSLz(49)
U3(4)
M12
U3(5)
Order
360 = 8.9.5
504 = 8.9-7
2520 = 8.9-5-7
4080 = 16.3.5.17
5616 = 16.27.13
6048 = 32.27.7
7800 = 8.3.25.13
7920 = 16.9.5.11
9828 = 4.27.7.13
20 160 = 64.9-5-7
20 160 = 64.9.5-7
25920 = 64.8105
29120 = 64.5.7.13
32736 = 32.5.7.31
58 800 = 16.3.25.49
62 400 = 64.3.25.13
95 040 = 64.27.5.11
126000 = 16.9.125*7
Group type
Janko group
A9
PSLz(64)
PSLz(81)
PSL3(5)
M2z
Newgroup
PSLz(121)
PsL&!5)
SP4(4)
Order
175560 = 8*3*5.7.11*19
181 440 = 64.81.5.7
262 080 = 64.9.5.7.13
265680 = 16.81.5.41
372000 = 32.3.125.31
443 520 = 128.9.5.7.11
604800 = 128.27.25.7
885 720 = 8.3.5.121.61
976 500 = 4.9.125.7.31
979200 = 256.9.25-17
3. Known results used. The known results used will be numbered for
references later in the paper.
(1) A paper, as yet unpublished, by John Thompson [33] determines the
minimal simple groups. These are among
PSL2(p), p prime, p 3 5
PSL2(2p) p prime
PSL2(3p) p prime
PSL3(3)
Su(2P) p an odd prime.
Not all of these are minimal. In particular any one of these whose order is a
multiple of 60 contains PSL2(5) = A5 of order 60.
(2) Daniel Gorenstein and John Walter [22] have shown that a simple
group with a dihedral Sylow 2-subgroup is necessarily a group PSLz(q), q
odd, or the group A7. In particular if the order of the group is not divisible
by 8 then it is divisible by 4 and is a group PSL,(q) with q E 3 or 5 (mod 8).
(3) Daniel Gorenstein [21] has shown that if a Sylow 2-subgroup of the
simple group G is Abelian, and if the centralizer of every involution is solvable,
then G is one of PSLz(q) where q = 3 or 5 (mod 8), q > 5 or q = 2”,
n * 2.
(4) Richard Brauer and Michio Suzuki [ll] have shown that a Sylow
2-subgroup of a simple group cannot be a quaternion group or generalized
quaternion group.
(5) It has been shown by J. S. Brodkey [13] that if a Sylow subgroup P of
a group G is Abelian then there exist two Sylow subgroups whose intersection
is the intersection of all of them. In particular if any two Sylow p-subgroups
have a non-trivial intersection, then the intersection of all of them is
a non-trivial group, necessarily a normal subgroup of G. Hence if a simple
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