Note on the equation s₁s₂ = s² ₂s²1, s₁ and s₂ being operators of a finite group.

Abstract

In 1878 Cayley*) proved that a group which is generated by two operators (s l9 s z ) satisfying the condition s l s 2 = slsl cannot have the property that each of its operators may be represented in the form sgsf. Recently Mr. Netto**} considered the same equation and proved, among other things, that s l and s 2 must either be of the same order, or the order of one of these operators is twice the order of the other. This interesting result may be stated more generally äs follows: // the operators s l and s 2 satisfy the equation s l s^^=s\s\^ then either the Orders of s l and s^ are equal to each other, or the order of one of these operators is an odd number and the order of the other is twice this odd number. Mr. Netto also determined the Orders of four groups which result by assigning certain small values to the Orders of s l and $ 2 · The main objects of the present note are to determine these four groups completely and to exhibit more fully the role of the equation given in the heading äs regards some important groups of finite order.Any operator and its inverse may be used for s ± and $ 2 respectively since such operators satisfy the equation Si^ -5 ! 5 !? an( l> ^ s i an d $2 are commutative, each one of them is the inverse of the other. In other words, if the group (G) generated by Sj^ and s z is abelian, it is also cyclic and 8 l9 *t represent respectively a generator of this cyclic group and the