THE HYPERORTHOGONAL AND RELATED LINEAR GROUPS. 129 



141. If r is not divisible by p and if r=^=p s -\-l, the structure 

 of the largest linear homogeneous group leaving O r (r > 2) invariant 

 is now evident. Indeed, the group has as a self - conjugate subgroup 

 the commutative group of the substitutions 



< = & (*'=-!, ...,w) [ = !], 



the quotient -group being the symmetric group on the m letters |,-. 



142. Theorem. - - Ike structure of the linear group in the GF[p n ~\ 

 which is defined by the absolute invariant <t> r , r=j?* + l>2, results 

 immediately from the structures of the groups in the G-F[p* s ] defined 

 by absolute invariants of the type 



For the case r =jp*+ 1, the conditions that S shall leave r in- 

 variant may be derived as special cases of 115) and 116), but are 

 given by inspection from the identity, 



By either method, the conditions in question are seen to be 

 117) V+i=^ 0' -!,...,), 



118) 



By 139, the inverse of 5 has the form 



By the same rule, the inverse of the latter substitution is 



. . .. 



3 



Hence this substitution must be identical with S. Hence 

 119) ; 



The determinant of S" 1 is 



Ja/f = #|- ,, 



DlCKSON, Linear Groups. 



j -!,...,). 



