COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

180
Degrc
-
No. Order t N (-63 Generators f Comments
- - ---
16 8 576 e.a.
17
18
19
20
9 960 e.a.
10 960 2 e.a.
11 960 2 e.a.
12 1152 e.a
13 1920 e.a.
14 1920 2 e.a.
15 2880 2 e.a.
16 5760 2 e.a.
17 5760 2p e.a.
18 11520 2p e.a.
19 40320 3 e.a.
20 ‘22560 3 e.a.
21 $16! 14~ -
22 16! 16 6G21
1 17 -
2 34 17Gl
3 68 17Gl
4 136 17Gl
5 272 2 17Gl
6 4080 3 -
7 8160 3 17G6
8 16320 3 17G6
9 $17! 15p -
10 17! 17 17G9
1 2448 2p -
2 4896 3 18Gl
3 i-18! 16p -
4 18! 18 18G3
1 19 -
2 38 19Gl
3 57 19Gl
4 114 19Gl
5 171 19Gl
6 342 2 19Gl
7 ;. 19! 171 -
8 19! 19 19G7
1 3420 2p -
2 6840 3 20Gl
3 +.20! 18~ -
4 20! 20 20G3
-
Charles C. Sims
TABLE 1 (continued)
15Gl
15G2
15G3
15G4
15G5
15G6
16G3
16G6
6GlC
6G21
6622
17G4
17G5
17G9
7GlC
18G3
18G4
19G5
19G6
19G7
19G8
bl5, %6.&. g1m
k,Lh,hl,
a153 a16, h6
wa5, a16, cl6
a15, alBr f 16
b157 a16y fiS, b-16,
h 1 6
a15, al,, ~1~
al51 al67 k16 a16e15, %6, f 16
a15e15p %Y k 16
a,,, hi, %6
%5, c15, %6
b15, d15, al6
d16, e15; aI6
al7
a17, b?,
a17, b4l7
a17, bf7
a179 b17
wk a16, cl7
wh a16, d6, cl7
a15e15y %6r e16Y '17
a17, %, a18
aI,, b17, al8
al8
alsl %
a197 b:,
b
al,, 19
a19, bf, agO
al8, b9, %O
-
-
-
-
-
-
-
-
-
-
-
A 16
S 16
PSL(2, 16)
A 17
S
I&(2, 17)
PGL(2, 17)
A 18
S 18
A 19
s
P&(2,19)
PGL(2, 19)
40
S 20
5
6
7
8
Computational methods for permutation groups
a = (1,2, 3,4, 5)
b = (1) (2, 3, 534)
a = (1,6) (2) (3,4) (5)
a = (1,2, 3,4,5, 6,7)
b = (1) (2,4,3,7,5,6)
c = (1) (2,3) (4,7) (5) (6)
a = Cl,81 (234) (3, 7) (5, 6)
b = Cl,41 (2,8) (395) (67)
c = (1, 7) (2, 5) (3, 8) (4, 6)
d = (1, 8) (2, 7) (394) (5, 6)
TABLE 2. Generating Permutations
~=(1,2,3)(4,5,6)(7,8,9)
b = (1, 4, 7) (2, 5, 8) (3, 6, 9)
c = (1) (2, 6, 4, 9, 3, 8, 7, 5)
d = (1) (2) (3) (4, 7) (5, 8) (6, 9)
e = (1) (2, 4, 9) (3, 7, 5) (6) (8)
f = (1) (2, 7) (3, 6) (4, 5) (8, 9)
a = (1, 8) (2, 5, 6, 3) (4, 9, 7, 10)
b = (1, 5, 7) (2, 9, 4) (3, 8, 10) (6)
c = (1, 10) (2) (3) (4, 7) (5, 6) (8, 9)
Permutations
a = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11)
b = (1) (2, 3, 5, 9, 6, 11, 10, 8, 4, 7)
c = (1, 11) (2, 7) (3, 5) (4, 6) (8) (9) (10)
3 = (1) (2) (3) (4, 8) (5, 9) 6 7) (10, 11)
z = (1, 12) (2, 11) (3, 6) (4, 8) (5, 9) (7, 10)
5 = (1) (2, 5) (3, 6) (4, 7) (8) (9) (10) (11, 12)
: = (1) (2) (3) (4, 7) (5, 8) (6, 9) (10) (11, 12)
z = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13)
i = (1) (2, 3, 5, 9, 4, 7, 13, 12, 10, 6, 11, 8)
: = (1) (2, 3) (4) (5, 10) (6) (7, 11) (8) (9, 12) (13)
I = (1, 14) (2, 13) (3, 7) (4, 5) (6) (8, 12) (9) (10, 11)
I = (1, 15, 7, 5, 12) (2, 9, 13, 14, 8) (3, 6, 10, 11, 4)
1 = (1, 4, 5) (2, 8, 10) (3, 12, 15) (6, 13, 11) (7, 9, 14)
: = (1, 7) (2, 11) (3, 12) (4, 13) (5, 10) (6) (8, 14) (9) (15)
! = (1, 9, 5, 14, 13, 2, 6) (3, 15, 4, 7, 8, 12, 11) (10)
t = (1, 3,2) (4, 8, 12) (5, 11, 14) (6, 9, 15) (7, 10, 13)
181
[continued on p. I82/