COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

146 Marshall Hall Jr. Simple groups of order less than one million 147
It is easy to find that if N(2)/C(2) is of order 3 then N(2) has a factor group
of order 2, and so by Grim’s theorem G has a normal subgroup of index 2.
Hence if g is divisible by 8 but not by 16, we may restrict our search to
orders which are also multiples of 7. We have incidentally proved the
uniqueness of the groups PSL2(q) as the only simple groups of their order
when g is divisible by 8 but not 16 or 7. In our list then there is a unique
simple group of each of the orders 360, 7800, 885 720, as well as orders
multiples of 4 but not 8, namely 9828 and 976 500, in each case the appropriate
PSLz(q).
A simple group G which is not minimal will contain some proper subgroup
which is not solvable and so has a simple group as a composition
factor. Hence G has a subgroup H and H has a normal subgroup K (possibly
K = 1) such that H/K is a simple group. We call a factor group of a
subgroup a section. Hence by John Thompson (1) we may confine our
search to groups which have one of the simple groups listed as a section.
Hereif[G:K]=tand [K: l]=k,andH/K=swehaveg=tsk.If
the simple group H/K is PSL2(p) for p == 41 then as g < 1,000,000,
s z= IPSL2(41)j = 34 440 it follows that tk -= 30. Then [G : H] = t .C 30
and so G has a permutation representation on t letters. But as p 2 41 G
cannot represent an element of order p faithfully on less than 30 letters.
Thus we may exclude as a section PSL&) with p z= 41. If PSLz(37) of
order 25,308 is a section, then tk < 39, and since G is represented as a
permutation group on t letters and contains an element of order 37, then
k = 1, t =Z 39. Here H = G1 is the subgroup of G fixing a letter and, as
PSL2(37) does not (from its character table) have a permutation representation
on less than 38 letters, it follows that t = 39. Then the order of
G is 25 308.39 = 987012 and by Gorenstein and Walter must be PSLz(q)
with q = 3, 5 (mod 8) which it is not. If the Suzuki group Su(8) of order
29 120 is a section of G, then tk < 34. Now Su(8) has as rational characters
the identical character, one of degree 64 and another of degree 91. It has
two algebraically conjugate characters of degree 14 both of which take
the value - 1 on elements of order 5, three algebraic conjugates of degree
35, and three algebraic conjugates of degree 65. From this it easily follows
that Su(8) has no subgroup of index less than 65, and this of course corresponds
to its representation as a doubly transitive group on 65 letters.
Since Su(8) has an element of order 13 we must have t Z= 14, and as tk < 34
then k = 1 or 2. With tk < 34 and either k = 1 or k = 2, the representation
of G on 34 or fewer letters with G either Su(8) or the extension of
Su(8) by a center of order 2 corresponds to a subgroup of Su(8) of index
less than 65, which is a conflict. Hence no simple G has Su(8) as a section.
PSL2(32) is of order 32 736 = 32.3 * 11.31. If PSL2(32) is a section of G
then tk < 30, clearly a conflict as we cannot represent an element of order
31 on 30 or fewer letters.
Having eliminated the groups above as sections of G, and having shown
easily that a section of a section is again a section, we have from John
Thompson’s results in (1) that one of the following minimal simple
groups is a section of G.
Group Order
P&(5)
60 = 4.3.5
P&(7)
168 = 8.3.7
PSLz@) 504 = 8.9.7
PSLz(12) 1092 = 4.3.7.13
PSL2(17) 2448 = 16~9.17
P&(3) 5616 = 16.27.13
PSL2(23) 6072 = 8.3.11.23
PSL,(27) 9828 = 4.27.7.13
From Gorenstein’s result (3) it follows that if the order of G is a multiple
of 8 but not 16, then the centralizer of an involution t is not solvable and
so H = COW/( z > is a non-solvable group of order a multiple of 4 but
not 8. Hence H contains as a section one of the groups PSL2(5), PSLz(13)
or PSLz(27) and so the order of G is a multiple of 8 * 3 * 5 -7 = 840 or of
8.3.7.13 = 2184.
On the basis of the above information we may divide the orders to
be examined into seven lists, the orders being multiples of particular
numbers.
Form of g Number of orders
A 16*3*5m = 240m 4166
B 16.3.7m = 336m, m $ O(5) 2381
C 8.3.5*7m = 840m, m odd 595
D 8.3.7.13m = 2184m, m odd, $ O(5) 183
E 16.9.17m = 2448m, m $ O(5), $ O(7) 280
F 16+27*13m = 5616m, m $ O(5), $ O(7) 124
G 16.3.11.23m = 12144m, m $ O(5), $ O(7) 57
Total number of orders 7786
As we have already remarked, if g is divisible by 8 but not 16 then g
is also divisible by 7 and from the Gorenstein result (3) G contains PSL2(5),
PSL2(13) or PSL2(27) as a section and so g is divisible by 5 or 13, giving
lists C and D. For multiples of 16, g is certainly a multiple of 3. If g is a
multiple of 5 or 7 it is included in lists A or B. If g is not a multiple of 5 or
7 it contains PSLz(17), PSL3(3) or PSL2(23) as a section and so is listed in
list E, F, or G. There are a few duplications between lists E, F, and G, but