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VII. On the Theori/ of Groups as depending on the Symbolic 

 Equation ^" = 1.— Part III * By A. Cayley, Esq.\ 



THE following is, I believe, a complete enumeration of the 

 groups of 8 : — 

 I. 1, a, a^ «3, «^ «^ ««,«' («« = !). 

 II. 1, u, ««, u^, 13, /3u, ^«2, y8«3 (a4= 1, ;32= 1, ccl3=l3u). 



III. 1, a, a^, oc^, /3, y8«, /3oc\ l3u^ {u'^= 1, /3^= 1, a/S = ySa«). 



IV. 1, «, «^ «^ /S, /S«, /3a2, /3«3 («4=1, ^2 = «^2, ufi^l3a% 

 V. 1, «, ^, yS«, 7, 7«, 7/3, 7^« («^= 1, ^2=1, y2= 1. 



«;S = /3«, «7 = 7a, /37 = 7/3). 



That the groups are really distinct is perhaps most readily 

 seen by writing down the indices of the different terms of each 

 group ; these are — 



I. 1, 8, 4, 8, 2, 8, 4, 8. 



II. 1, 4, 3, 4, 2, 4, 2, 4. 



III. 1, 4, 2, 4, 2, 2, 2, 2. 



IV. 1, 4, 2, 4, 4, 4, 4, 4. 

 V. 1, 2, 2, 2, 2, 2, 2, 2. 



It will be presently seen why there is no group where the 

 symbols a, /8 are such that «"*=!, /S^=l, a/3=/3ct^. A group 

 which presents itself for consideration is 



1, «, «2, a^^ /3, y8«, /3«^ /3«3 («4=i, /3^=«2, «/e=^«); 

 but the indices of the different ternis of this group are 

 1, 4, 2, 4, 2, 4, 2, 4, 



and if we write /3a = y, then we find 7^ =/3a/3a=/3/3a« = «"* = !, 

 «7=a/3a=/8aa=y«; and the group is 



1, a, «2, «3, 7, 7«, ya^^ ya^ («'' = 1, 7^=1, «y = 7«), 



which is the group II. 



The group IV. is a remarkable one ; it appears to arise from 

 the circumstance that the factors 2 and 4 of the number 8 are 

 not prime to each other ; this can only happen when the num- 

 ber which denotes the order of the group contains a square factor. 

 But the nature of the group in question will be better understood 

 by presenting it under a different form. In fact, if we write 

 /3«3 = y, u^=^=b, then we find «3 = ;&a, /3ot' = d^, fiu = 5,y, 

 and the group will be 



1, a, fi, y, ^, 5«, h^, ^7, 



* See Phil. Mag. vol. vii. (1854) pp. 40, 408. 

 t Communieated by the Author. 



