340 REV, T. V. KIRKMAN ON THE THEORY Of 



We must have either 



RG=„G, 

 or 



KG -\= „G, 



where =1= „ denies algebraic identity. 



If 



I^G =1=^ Jj, 



E, must exchange an element x^ which carries an exponent 

 /8 in G for an element carrying a different exponent. Let 

 us suppose that R has the substitutions 



where a?^ is an element carrying in Pj and in G a diflPerent 

 exponent from that of <2?„. 



The effect of the operation RG is to change x^ into Xf, 

 in every vertical row in which a\ appears in the function # 

 written in a column, and to disturb in the same vertical 

 rows no elements which occupy in R their natural posi- 

 tions. 



We have the two conditions 



RGH=«G 

 RGIl-^ = „G. 

 Then the derangement of RG by R~^ exactly compensates 

 the algebraic disturbance produced by the operation RG. 

 Now this derangement affects entire vertical rows of RG 

 and can affect no row standing under an element of unity 

 not displaced in R"^, while the operation RG disturbs ie^ 

 in every row in which it appears. 



It is then impossible that this compensation of algebraic 

 disturbance can take place, and that 



RG =H ,G. 

 Wherefore 



RG=„G 

 if 



G' = ^RGR" = Jj } 



that iS; G gives the same algebraic function with its 



