On Finite Groups with Two Independent Generators: II

J. Douglas
1951 Proceedings of the National Academy of Sciences of the United States of America  
The present note is a direct continuation of Note I of the same title,' with which we must suppose the reader acquainted, and to which we shall therefore refer quite freely. 8. Order of Cycles.-The following, generalizing Theorem VII, limits severely the nature of a special substitution. THEOREM XII. Let 0 be a special substitution on [m] and a, b any two numerals of this range. Then if (a, m) (b, M),2 the order of the cycle containing b divides the order of the cycle containing a. It follows
more » ... at if (a, m) = (b, m), then a and b belong to cycles of the same order. Thus all the numerals prime to m belong to cycles of the same (maximum) order, which order is divisible by that of every cycle. For illustrative material pertaining to this and subsequent theorems, we may use the following special substitutions, which are listed with their derivatives.
doi:10.1073/pnas.37.10.677 pmid:16578401 pmcid:PMC1063445 fatcat:tbgflipsc5bbhnf5p6p4nzzupi