COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

166
Marshall Hall Jr.
Since t moves all letters it must interchange the orbits of odd length. In
particular if t = (03, Ai) then Ai is one of the numbers 51,. . . ,57,65,. . , ,71,
72,. . ., 78. As tat = a-l,
where(-,Ai,-,Ai...)
(02,03,04,05,06,07,08) (6.15)
(-, Ai, -, Ai, . . . . . . . a)
is in some order (51, 57, 56, 55, 54, 53, 52),
(65,71,70,69,68,67,66) or (72,78,77,76,75,74,73) and also b2dl = zi .
0 i
We have
b2a4 = (OO)(Ol, 13, 35,26,04, 24, 15, 30)(02, 20,21, 14)
(03, 10, 18, 19, 33, 05, 11, 32)(06, 36, 16, 17)
(07, 23, 34, 22,09,27,08, 25)( 12, 31, 29,28)
(37, 41, 60, 88, 56, 53, 90, 61)(38, 71, 66, 40, 44, 62, 59, 50)
(39, 79)(42, 57, 73, 78, 52,43,45,49)(46, 51, 48,93, 84, 89, 81, 99)
(47, 72)(54, 65, 55, 85, 67, 96, 70, 80)(58, 69, 77, 86, 83, 98, 95, 75)
(63,76,97, 94, 82, 92, 74, 68)(64)(87,91)
(6.16)
Here Ai = 73, Aj = 78 is the only pair from the 21 orbit of (a, d) in
Ai
the 63 orbit of H satisfying b2d =
0
A. and a-l = (. . . Ai,-,Aj. . .).
Hence from (6.13) we must have t = (03,‘73). The equation (6.13) and the
relations tat = a-l, tc = ct now completely determine t:
t = (00,01)(02, 74)(03, 73)(04, 72)(05, 78)(06, 77)(07, 76)(08, 75)(09, 34)
(10, 33)(11, 32)(12, 31)(13, 30)(14, 36)(15, 35)(16, 71)(17,70)(18, 69)
(19, 68)(20, 67)(21, 66)(22, 65)(23, 53)(24, 52)(25, 51)(26, 57)(27, 56)
(28, 55)(29, 54)(37,91)(38, 90)(39, 89)(40, 88)(41, 87)(42, 86)(43, 92)
(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(58, 85)(59, 84)
(6.17)
(60, 83)(61, 82)(62, 81)(63, 80)(64, 79).
We now have a permutation group G on 100 letters 00, 01,. . . ,99,
G = (a, b, an t) d a group
H = (a, b) fixing 00 where it is known that
H is the simple group of order 6048. Let M = Gee be the subgroup of
G fixing 00. It is well known that M is generated by the 300 elements
xia Xp-“, xtb z-l, xit Xif-ly
(6.18)
, i=OO,..., 99, are coset representatives of M in
G and 7 = xj is the coset representative of My = MXj. Here, with the help
Simple groups of order less than one million 167
of Mr. Peter Swinnerton-Dyer and the Titan computer at the Cambridge
University Mathematical Laboratory, it was shown that each of the 300
permutations in (6.18) lies in H = (a, b). Thus M C H, and so M = H,
whence Goo = H and G is of order 604,800.
It remains to be shown that G is simple. In G the normafizer of the group
(a) contains the element t interchanging 00 and 01. As these are the only
letters fixed by (a) it follows that [N&(a))] = 2. As INH((a))I = 21 it
follows that IN&(a))/ = 42 and (a) = S(7) is its own centralizer in G.
In a chief series for G one of the factors has order a multiple of 6048.
If this is not a minimal normal subgroup, then a minimal normal subgroup
K has order 2, 4, 5, or 25. But then an S(7) normalizing K must also centralize
K, which is false since S(7) is its own centralizer. Hence a minimal
normal subgroup K has order a multiple of 6048 and also 14,400 since
it must contain all 14,400 S(7)‘s. Thus either [GX] = 2 or G = K and
G is simple. If [G:K] = 2 then N,(7) is of order 21 and so tcj K but HE K.
But mapping G/K onto the group + 1, - 1 we map H- + 1, t - - 1 and
this conflicts with the relation (6.12). Hence G is simple.
As this is written some questions remain unanswered. The original
character table given by Janko has been shown by Walter Feit to be in
error. Janko has made corrections to his table. The character table of the
group constructed here has not yet been calculated. And it has not been
established that there is a unique simple group or order 604,800.
[Added in proof by the Editor: The uniqueness of the simple group of
order 604,800 has since been established ; see Marshall Hall Jr. and David
Wales: The simple group of order 604 800, J. Algebra 9 (1968), 417-450.]
REFERENCES
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groups of finite order. Ann. of Math. 42 (1941). 926-935.
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225-255.
8. R. BRAUER and K. A. FOWLER: On groups of even order. Ann. of Math. 62 (1955),
5655.583.
9. R. BRAUER and C. NESBITT: On the modular characters of groups. Ann. of Math. 42
(1941), 556-590.
10. R. BRAUER and W. F. REYNOLDS: On a problem of E. Artin. Ann. of Math. 68 (1958).
713-720.
CPA 12