LINEAR HOMOGENEOUS GROUP IN THE GF[2n] etc. 201 



We study the group ft of 2w-ary linear substitutions in the 6rF[2 n ], 



192) S: 



(i = l, ..., w) 



which leave fa absolutely invariant. The conditions upon the coeffi- 

 cients of S are the Abelian relations 1 ) 76), for ft = 1, together with 



193) 



u=i 



Since $ must be an Abelian substitution in the 6r.F[2 n ], its reciprocal 

 is obtained by replacing a ih fa, y ih d {j by respectively dj t , ft-,-, y jiy 

 dji. Writing for S~ l the conditions 76) and 193), we obtain the 

 equivalent set of conditions 78), for ft = 1, and 



194) 



t=i 



TO 



(j 



Among the simplest substitutions leaving fa invariant occur 



(i, j < m if A = 



J* f f fc i 1 "* "" ^ f* *\ ' 



*9ffl *{Wl J '/771 ^~"" ^771 I " *|77l \ 



which reduce, when A = 0, to the JV^,*, R^j,*, etc., defined in 114. 

 According as A = or A = A', ft is called #&e /?rs^ or fe second 

 hypoabelian group*). The name arises from the fact that ft is a 

 subgroup of the special Abelicm group SA(2m y 2 ra ). 



1) This also follows from the fact that the invariance of fa implies that 



m 



of its polar. Hence, ifp = 2, G^ leaves invariant^ (^ti 7 ?2~H ^t2^i)' wliere i,-i, 



t==i 

 ^ fl and l^.g, rj is are sets of cogredient variables. 



2) For the case n = 1 , these groups were studied at length by Jordan, 

 Traite des substitutions, pp.195 213 and p. 440. For general w, they were 

 set up and investigated by the author in the papers, Quarterly Journal, 1898, 

 pp. 116; Bulletin of the Amer. Math. Soc., 1898, pp. 496 510; Proceed. Lond. 

 Math. Soc., vol. 30, pp. 70 98; American Journal, 1899, pp. 222 243. 



