Groups generated by two operators (π β,π β) satisfying the equation π βπ Β²β=π βπ Β²β
G. A. Miller Β· Transactions of the American Mathematical Society Β· 1910
In 1878 Cayley considered the groups generated by two operators (sx,.s2)satisfying the equation s, s2 = s\ s\ and observed the interesting theorem that it is not possible to represent all the operators of such a group in the form s" s* except in the special trivial case in which the group is cyclic.fThis theorem is of historic interest as it is one of the earliest theorems relating to a general category of abstract groups.More recently Netto published a few additional general results which may be deduced from this equation and he also determined the possible groups when the orders of sx, s2 are both less than 6.J The results obtained by Netto were extended by the author of the present paper, mainly by means of special considerations, as regards the possible groups in which the two generating operators are of order 6. Β§For convenience the equation under consideration is written, in the present paper, in the form s,s\ = s2s\, and a number of new general results are deduced from it.By means of these the known results are obtained much more easily than in the earlier papers.It is also proved that the two equations s\ = 1, sxs\ = s2s\ imply that s\ = 1 and hence either the first or the second of the three generational relations s\ = 1, s\ = 1, sx s2 = s2 s2 given by Netto in the article cited above is redundant.This is of interest since it proves that the non-cyclic group of order 55 is the only non-abelian group which can be generated by two operators satisfying the two conditions sj = 1, *,s2 = s2s2x and hence it establishes contact between the present paper and the one devoted to the " Finite groups which may be defined by two operators satisfying two conditions."|| Among the other results the following are perhaps of most interest.