COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

38
Repeating the process,
Finally, from the tables,
Therefore
and hence
C. M. Campbell
a.a, . . . a, = Wr-lWr--2 . . . WlW, . . . W.+.&+~.
/a+1 = Ws+l.B.
a.a, . . . a, = W,-IWr--2 . . . WIWP . . . Wsfl,
a.a, . . . us = W./T
The Todd-Coxeter process leads to an enumeration table and from the
modification and our theorem (withp = 1) we obtain a table carrying additional
information (see Example 1).
Examples. In [2] an algorithmic proof is given to show that the two
relations
RS= = S3R, SR2 = R3S
imply that R = S = E, where Eis the identity. This has been generalized by
Benson and Mendelsohn [4] who show that the two relations
RS”-1 = S”R, sR”-1 = R”s
again imply that R = S = E.
We consider two examples that arise from the previous two.
EXAMPLE 1. Let G = {R, S, T, U} be subject to the relations RS = SST,
ST = TSU, TU = U2R, UR = R%. Then G is cyclic of order 5.
Proof. RS = S2T (1)
ST=T2U (2)
TU = U2R (3)
UR = R=S (4)
ST2U = RS (5) from (11, (2)
STU=R (6) from (3), (4), (5)
where, as before, (-) denotes the inverse
SU = RST (7) from (3), (6)
S3T = E (81 from Cl), (2), (6)
SRS = E (9) from (0, (8)
U2 = R (10) from (4), (9)
ST = EzU (11) from (6), (10)
s =@ (12) from (7), (11)
IJ = p (13) from (4), (12)
T=i? (14) from (1 I), (12), (13)
R3 = E (15) from (10)
Some examples using coset enumeration 39
This algebraic proof follows algorithmically from the modified Todd-
Coxeter process with the following enumeration and information tables.
New information is found from the underlined positions in the tables in the
order numbered.
R Slss S TUFT TUi?DU
1 1 2(2)3 2 1 1 2 5 1 1 4 1
2 (7) 1 3 2 2 3 2 (3) 5 2 2 5 1 1 5(5)2
3 3 3 (6) 1 5. 2 3 3 2 3
4 2 3 4 4 4 4 4
5 5 5 5 5 2 4 1(4)5
coset Representative R S T u
-
1 4(1)2 1 1 1
2 5 2
3 3
4(8)1 1 2 4
5 1 1 5
1 = lRj l=E l.R = R.l 1.5’ = E.2 l.U = E.4
2 = i.s- 2=s 2.S = E.3 2.T = ES 2.u = R.5
3 = 2.s 3 = s2 3.T = R.2
4 = l.u 4=u 4.R = Rz.2 4.U = R.l
5 = 2.T 5 = ST 5.u = a.1
From the positions numbered 3, 6, 7 we have, using our theorem, the
additional information
5.TU = R.2,
3.ST = E.1,
2.RS = E.1.
In the above tables 4. U = R.l and 5. U = i?. 1, which implies that 4 and 5
are the same coset, and in terms of coset representatives 5 = R2.4. Replacing
5 by 7(i2.4 in the information tables gives
R S T U
l.R = R.l l.S = E.2 l.u = E.4
2.S = E.3 2.T = i?.4 2.u = IF.4
3.T = R.2
4.R = R=.2 4.U = R.l
From these tables 1. U = E.4 and 2. U = W3.4, and it now follows that
1 and 2 are the same coset. Repeating the process as before leads finally to
complete collapse.
CPA 4