COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

40 C. M. Campbell Some examples using coset enumeration 41
The new information 4.R = 2 and 3.T = 2 reduce to equations (1) and
(4) but from the new information 5.TU = 2 equation (5) is obtained. In the
first of the two calculations below we work with the coset representative as
an integer and in the second we think of the coset representatives as a word
in the group.
5.TU = 5.TST ST.TU = ST.TST from (2)
= E2.ST = ES.ST
= EE3.T = EES=.T
= EER.2 = EER.S
5.TU = R.2 = R.S from (1)
This is equation (5). In a similar manner we obtain equations (6)-(10).
Equation (11) comes from the first coincidence when cosets 4 and 5 are
identified.
5 = a1.o from 5.u = R.1
= RR4.ui7 from 4.U = R.l
= f72.4,
or, in terms of coset representatives,
ST = iZE.D
= muuu
= i?=u,
from (6)
from (10)
and this is equation (11). From the other coincidences we obtain equations
(12)-(14).
EXAMPLE 2. The relations SR2 = RSRS and RS2 = SRSR imply that
R=S=E.
Proof. The proof is again obtained algorithmically as in Example 1.
SR= = RSRS (1)
RS= = SRSR (2)
SR3 = R=S= (3) from Cl), (2)
S=RSR = RSR=S (4) from (11, (2)
S3 = j&‘9R= (5) from (11, (3)
SR=SRS = RS2R (6) from Cl), (3)
S=R=S = S=R= (7) from (2), (3), (5), (6)
whence S = E and R = E.
The following question now arises. Given SR” = RnmlSRS and RS” =
= Sn-IRSR, do these relations imply R = S = E? (True for n = 1, 2.)
One further example is the following: show that the group generated by five
generators a, b, c, d, e subject only to the relations ab = c, bc = d, cd = e,
de = a, ea = b, is cyclic of order 11. This problem was discussed in the
American Mathematical Monthly [S].
REFERENCES
1. H. S. M. COXETER and W. 0. J. MOSER: Generators and Relations for Discrete Groups.
(Springer, Berlin, 1965).
2. C. M. CAMPBELL: Dissertation, McGill University, 1965.
3. W. 0. J. MOSER: The Todd-Coxeter and Reidemeister-Schreier methods. Lecture
delivered at the Conference on Computational Problems in Abstract Algebra held
at Oxford, 1967.
J4. C. T. BENSON and N. S. M ENDELSOHN: A calculus for a certain class of word problems
in groups, J. Combinatorial Theory 1 (1966), 202-208.
5. Problem 5327. American Math. Monthly 74 (1967), 91-93.