308 CHAPTER XV. 



A (2m, p n )l ( 



where p n > 3 if m = 1, p w > 2 if w = 2, and a = 1 if^) = 2, a = 2 

 ifp>2. 



F0(2m + 1, # w ): y O n(2m) - 1) jp-P*"- 

 (p zn T)p n 



where p > 2 and, for w = 1 7 p w > 3. 



where j) > 2 and m > 2 7 while = + 1 according as p n = 4Z + 1. 



, 



where ^) > 2 and m > 1 ? = 1 according as p n = 4? + 1. 

 FH(2m, 2): (2 MW 



where w > 2. 



SH(2m, 2 n ): (2 ram + 1) (2 2w ( m - 1 ) - 1) 2 2 *(- 1 ) . . . (2 2 w - 1) 2 2 % m 



In addition to these systems may be added the cyclic groups of 

 prime order and the alternating group on n > 4 letters. 



288. Between certain of the above groups there exists holoedric 

 isomorphism, a relation indicated by the symbol ~. For p > 2 7 the 

 following isomorphisms were established in 178, 187 190, 

 197_198: 



,p n }; F0(6,p n ) ~ LF(4=,p n ), for #= 4Z + 1; 

 w ) ; /S 0(6, p n ) ~ LJP(4, p n ), for ^ = 4? + 3; 



the latter holding also for jp n = 3, a case not treated in 197 198. 

 For any p, 



LF(2, p n ) ~ A(2, p n ) - H0(2, p* n ). 



For p = 2, it was shown in 198, 206, 207 that 



#(4,2*)~Z.F(2,2 2 ) 



By chapter XIII, 



, 2 2 ), Zr ^(4, 2) - G , iF(3, 2) - Z^(2, 7). 



8 1 



