482 Dr Burnside, Certain Simply-Transitive Permutation-Growps 



On Certain Simply-Transitive Permutation-Groups. By Dr W. 

 Burnside, Honorary Fellow of Pembroke College. 



[Received 19 September 1921.] 



In 1900 1 proved {Proc. L. M. S. vol. xxxiii, p. 177) that a simply- 

 transitive permutation-group of prime degree p must contain a 

 self-conjugate subgroup of prime order. In the second edition of 

 my Theory of Groups (1911) it was shown that a simply-transitive 

 permutation-group degree 2?"^ which contains a permutation P of 

 order f'^, necessarily has a self-conjugate subgroup containing 

 pp"^- , I ventured then to express an opinion that a similar result 

 was true for any simply-transitive permutation-group which con- 

 tained a transitive Abelian subgroup. Quite recently I have suc- 

 ceeded, with a single exception, in justifying this expression of 

 opinion in a remarkably simple way. 



Denote by x^^ ^ a set of mn variables, where the first suffix takes 

 all values from to m— 1, and the second all values from to w — 1. 



The permutation x' = x 



is a regular permutation M, of order m, in the 7nn variables, and 



is a regular permutation N, permutable with M. The two generate 

 a regular AbeKan group {M, N} simply-transitive in the mn 

 variables. 



If e, rj are primitive ?«th and nth. roots of unity, and if 



ab 



then Mi,,, = e%,; iVf,,, = V|.-,.-, 



so that the mn quantities i^^j are the reduced variables for the 

 Abehan group {M, N} and each gives a distinct representation of 

 the group. Also 



i,} i,j, a,b 



fit /tt/yy^ ^ • 



Suppose now that a simply- transitive group G in the mn x's 

 contains {M, N}. Since no irreducible representation of {31, N} 

 occurs more than once, no irreducible representation of G, when it 

 is completely reduced, can occur more than once; and therefore* 

 Go, the subgroup that leaves Xo,o unchanged has just one hnear 

 invariant in each irreducible representation of G. 



For each irreducible representation of G, the reduced variables 

 must be expressible as a set of |'s. Let 



^«i, h -' ^«2, ^2 ' • • • ' ^v, ^^L 

 be the reduced variables for an irreducible representation T. 



* Theory of Groups, Second Edition, p. 275. 



