Dr Burnside, A group of order 660. 247 



On the representation of the simple group of order 660 as a group 

 of linear substitutions on 5 symbols. By Dr W. Burnside, Honorary- 

 Fellow of Pembroke College. 



[Read 22 November 1920.] 



Except in the cases of two and of tbree variables, it is only few 

 groups of linear substitutions of finite order the forms of which 

 have been exhibited explicitly. This, it is hoped, will justify the 

 following calculations, which seem to offer one or two points of 

 interest. In particular the existence of a cubic three-spread in 

 space of four dimensions, which admits a group of 660 coUineations 

 into itself, is perhaps noteworthy. The well-known cubic three- 

 spread of Segre, defined by 



(/y I /y* .J_ 'y* I /y _' /y \o . rv" 3 /y 3 /y> 3 /yi 3 _^ /y 3 f\ 

 «/--Q n^ »^i \^ »^2 '^ 3 ^^ 4/ 1 2 'i 4 — ' 



admits a group of 720 coUineations. That which admits the group 

 of 660 coUineations is defined by 



The modular group, for j9 = 11, is a simple group of order 660. 

 Its characteristics have been calculated*, and it is known to admit 

 two representations as an irrational irreducible group on five 

 variables. It is proposed here to set up these two representations. 

 In one of them the multipliers of an operation P of order 11 are 

 a, a^, a*, a?, a^ ; where a is a primitive 1 1th root of unity. Moreover 

 the group contains an operation S of order 5 such that 



SPS--^ = P9. 

 Now when P has the above multipliers, there is only one representa- 

 tion of the cyclical group {S, P) as a group of degree 5 ; and by suit- 

 ably choosing the variables this can be brought to the form which 

 is generated by 



/y» — /vy 'V* — f^'^'V 'y — /^^^ 'V* — /^*^^y* ^y — /^"^y 



Q vi ri ^y — -y nf* — 'y •y — o^ ^y — o™ ^ — ^ 



The variables for the required irreducible representation V may 

 be therefore chosen so that V contains this subgroup. When the 

 methodf for finding the number of invariants of the nth degree 

 of a group is applied to V it is found that the group has one cubic 

 invariant. Now it is very easily verified that the only cubic invari- 

 ant for the above metacyclical group {>S, P) is 



/y £i/y [ ly Ji>y 1 /y fti^y —1— 'y ^'y \ T* ^/y 



and this must therefore be an invariant for F. 



* Burnside, Theory of Groups, p. 502. f loc. cit. p. 301. 



