Dr Burnside, Certain Simply-Transitive Permutation-Cfroups 483 



Since 7nn Xq, o = S |,_ j, the one linear invariant that Gq has in 

 r is ' ^'J 



Suppose now that 



1 = 1 



■^oi, 61 J "^02, f»2 ' • • • ' '^ap, bp ) 



is a set of variables which are permuted transitively by Gq. Then 



3=1 ■' •' 



is a linear invariant for Gq . 



Now 2^„.p.= i; Se-^t-7;-^^%,,, 



i = 1 i = 1 o,& 



and since the right-hand side is invariant for Gq which permutes 

 ^aj,bj (i = 1. 2, ... p), transitively, 



1 = 1 



is independent of _;'. 



p p 



Also wmS a-^.ft^S e«;"r?^-^'|„,„, 



j = i ^ -^ 3 = 1 



and since the only hnear invariant that Gq has in the symbols 

 $ai,Pi {i = 1,2, ... n) is their sum 



p 



y = l 

 is independent of *'. 



Moreover the immediately preceding result may be expressed 

 in the form that 



2 e"«"j yfi^i 

 i = l 



is independent oij. 

 Hence 



ti. p p y- 



2 S e'^i'^irf'fii = /x S e'^i'^irj^i^i = pi, e'^i'^iyfi^i. 

 i=lj=l i=l i=l 



Now 2 e"j"«7j^3^j 



i = l 



is the sum of the multipliers of the operation, M^iN^i is the irre- 

 ducible representation V. Hence in T 





for each i. 



