COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

162
Order of an element
Order of centralizer
Yl
Ya
W
~lfy'afyle = x
Marshall Hall Jr.
1 2 4 8 6 12 10 10 2 6 10 10
g 1920 96 8 24 12 10 10 240 12 20 20
1 1 1 1 1 1 1 1 1 1 1 1
36 4 40 1 1 -l -l 0 0 0 0
63 15 3 1 0 0 0 0 -1 -1 -1 -1
100 208222 0 0 0 0 0 0
3 3 5 5 5 5 15 15 7
1080 36 300 300 50 50 15 15 7
1 1 1 1 1 1 1 1 1
9 0 -4 -4 1 1 -1 -1 1
(6.1)
0 3 3 3 -2 -2 0 0 0
10 3 0 0 0 0 0 0 2
Under the assumption that x is the character of a group G of order
604,800 represented as a permutation group on 100 letters, the group G will
be constructed and it will be shown that G is simple.
The stabilizer of a letter G, is of order 6048. From the fact that an S(7)
is its own centralizer it is not difficult to show that G, must be the simple
group Us(3). As x is the sum of three irreducible characters of G it follows
that G, has precisely two orbits besides the fixed letter 01, and in D. G. Higman’s
terminology, G is a rank three group.
In GF(33 the mapping x-+x3 = 2 is an involutory automorphism. The
unitary group Us(3) consists of the linear transformations over Gt;(32) on
x, y, z leaving invariant for points A = (x, y, z) the metric
(A, A) = xx+yjJ+zz. (6.2)
A natural representation for H = Us(3) is as a permutation group on the
28 points A for which (A, A) = 0 considering these as points in the projective
plane PG(2, 33. These lie in sets of four on 63 lines and form a block
design D with parameters b = 63, v = 28, r = 9, k = 4, A = 1. Number the
points 1, . . ., 28 and the lines 37, . . . , 99 we have for the set of lines con-
taining the point 1
L 37. - 1, 2, 3, 4
L,: 1, 5, 18, 28
L51: 1, 6, 14, 23
L581 1, 7, 15, 22
L *
65. 1, 8, 19, 25 (6.3)
L72: 1, 9, 16, 27
L,: 1, 10, 17, 24
:
L,: 1, 11, 20, 21
L&l 1, 12, 13, 26
Simple groups of order less than one million 163
Here His generated by the permutations
a = (1, 5, 7, 3, 12,24, 11)(2,23, 4, 27, 13, 14,26)(6, 20, 18, 8, 25, 21,28)
(9, 10, 17, 15, 22, 16, 19),
b = (1)(2)(3, 4)(5, 17, 7, 16, 8, 20, 6, 13)(9, 19, 11, 14, 12, 18, 10, 15)
(21,23, 26, 28, 24, 22, 27, 25).
(6.4)
The Sylow 7-group (a) is normalized by the element c and N(7) = (a, c)
where
c = (1)(2,20, 10)(3, 11, 24)(4, 21, 17)(5, 7, 12)(6, 19, 27)(8, 16, 23)
(9, 14, 25)(13, 18, 15)(22, 26, 28). (6.5)
Here (c) is normalized by the involution d where
d = (1)(8)(19)(25)(2, 28)(3, 18)(4, 5)(6,27)(7, 17)(9, 14)(10,22)(11, 13)
(12, 21)(15, 24)(16, 23)(20, 26). (6.6)
We shall construct G as a permutation group on the 100 symbols OO,Ol, . . . ,
99 and H shall be the stabilizer G 00. From the character x of (6.1) we see
that a 7-element fixes exactly one of the 99 letters moved by H; thus of the
two orbits of H on the 99 letters one is a multiple of 7 in length, the other of
length congruent to one modulo 7. As both orbit lengths are divisors of
6048, the only possible lengths are 36 and 63. The 63 orbit for H will be on
the blocks of D numbered from 37 to 99. The 36 orbit will be on cosets of
the subgroup (a, d) of order 168 and isomorphic to PSL2(7), numbering
the cosets from 01 to 36. The permutations representing the elements
a, b, c, d are as follows:
a =
(00)(01)(02,03, 04, 05,06,07, 08)(09, 10, 11, 12, 13, 14, 15)
(16, 17, 18, 19, 20, 21, 22)(23,24, 25,26,27,28,29)
(30, 31,32,33,34, 35, 36)(37, 38, 39,40,41,42,43)
(44,45,46,47, 48,49, 50)(51, 52, 53, 54, 55, 56, 57)
(58, 59, 60, 61, 62, 63, 64)(65, 66, 67, 68, 69, 70, 71)
(72, 73, 74, 75, 76, 77,78)(79, 80, 81, 82, 83, 84, 85)
(86, 87, 88, 89,90,91, 92)(93, 94, 95, 96, 97, 98, 99)