COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

182 Charles C. Sims Computational methods for permutation groups 183
TABLE 2 (continued)
Degree Permutations
16 P = (1, 16) (2, 3) (4, 5) (6, 7) (8, 9) (10, 11) (12, 13) (14, 15)
b = (1, 3) (2, 16) (4, 6) (5, 7) (8, 10) (9, 11) (12, 14) (13, 15)
c = (1, 5) (2, 6) (3, 7) (4, 16) (8, 12) (9, 13) (10, 14) (11, 15)
d = (1, 9) (2, 10) (3, 11) (4, 12) (5, 13) (6, 14) (7, 15) (8, 16)
e = (1, 12, 7, 5) (2, 4, 13, 11) (3, 8, 10, 14) (6, 9) (15) (16)
f= (1, 3, 2) (4, 12, 8) (5, 15, 10) (6, 13, 11) (7, 14, 9) (16)
g = (1) (2, 3) (4) (5) (6, 7) (8, 12) (9, 13) (10, 15) (11, 14) (16)
h = (1, 15) (2, 12) (3) (4, 10) (5) (6) (7, 9) (8) (11) (13) (14) (16)
i = (1, 7) (2, 12) (3, 11) (4, 10) (5, 13) (6) (8) (9, 15) (14) (16)
j = (1, 14) (2, 13) (3) (4, 11) (5) ( 6) (7, 8) (9) (10) (12) (15) (16)
k = (1, 3) (2) (4, 8) (5, 11) (6, 10) (7, 9) (12) (13, 15) (14) (16)
17 a = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17)
b = (1) (2, 4, 10, 11, 14, 6, 16, 12, 17, 15, 9, 8, 5, 13, 3, 7)
c = (1) (2, 3) (4, 9) (5, 7) (6, 8) (10, 14) (11, 13) (12, 15) (16, 17)
18
19
20
a = (1, 18) (2) (3, 10) (4, 7) (5, 14) (6, 8) (9, 16) (11, 13) (12, 15) (17)
a = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19)
b = (1) (2, 3, 5, 9, 17, 14, 8, 15, 10, 19, 18, 16, 12, 4, 7, 13, 6, 11)
a = (1, 20) (2, 19) (3, 10) (4, 7) (5, 15) (6, 16) (8, 9) (11, 18) (12, 13) (14, 17)
is given in the form of two tables. In Table 1 the groups are listed together
with some facts about them. In Table 2 we give the generators for these
groups which are referred to in Table 1. The groups in Table 1 are listed
by degree and for a fixed degree by order. Beyond this the numbering is
arbitrary. The order of the group is listed as is the transitivity t, whenever
t s 1. If the group is t-fold primitive, this fact is indicated by the letter p
following the transitivity. Generating permutations are given for all groups
except the alternating and symmetric groups of each degree. The entries
in this column refer to permutations in Table 2. For example, a, denotes the
first permutation listed under degree 7 in Table 2. No attempt has been
made to give generating sets with the fewest possible elements. Permutations
in Table 2 should be considered defined on and fixing all integers
greater than the degree under which they are listed. A minus sign (-) in
the column headed f indicates that the group contains odd permutations.
Whenever the group G is doubly primitive so that the subgroup G,, fixing
the last integer on which G acts is a primitive group of degree IZ- 1, this
subgroup G, is given. The symbol 6G3, for example, refers to the third
group of degree 6. In all cases G,, is the actual group listed and not just
permutation isomorphic to it. Any primitive group G of degree n -K 60
has a unique minimal normal subgroup N. If n + 20, then N is simple
and primitive or elementary abelian. If N is primitive, it is listed, with a
dash (-) indicating that N = G. The letters e.a. mean that N is elementary
abelian and imprimitive. We note that for groups 1, 2, and 5 of degree
8 N = (aa, b,, cs), for groups 1 to 7 of degree 9 N = (as, bs), and
for groups 1 to 20 of degree 16 N = (a16, bls, ~16, dl+. Whenever the
group is abstractly isomorphic to a member of one of the families of groups
A,, S,, PSL(n, q), PGL (n, q), this fact is noted in the last column. In those
cases involving the groups As z PSL(2,4) z PSL(2,5), ,I$, % PGL(2, 5),
PSL (2, 7) 2 PSL (3, 2), As z PSL (2, 9) and As z PSL (4, 2), only one
of the two or three possible designations is listed.
Great care has been taken to ensure the accuracy of these tables. However,
the author would appreciate being informed of any errors that may be
found.
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CPA 13