Examples of Groups 

 with the multiplication table — 



The subgroups are- 



{i, m\, 1 1, r\, 1 1, mr, rm\, \i, mrm\. 



That these are the only subgroups is easily seen ; for either 

 in, r, or mrm, together with any second operation of the group, 

 will generate the entire group. 



If, in the triply-crossed hexagon 

 to the left, /denotes motion from end 

 to end of a full line, while d denotes 

 motion from end to end of a dotted 

 one, it is plain that d and/ generate 

 a group differing from the previous 

 one only in the substitution of d for 

 m and / for r. But the group fits 

 precisely as well the uncrossed hexa- 

 gon, just as the original group could 

 have been entirely expressed in terms 

 of mrm and m instead of r and m* 



*If n denotes mrm, the multiplication table in terms of n and m will 

 come from that in terms of r and m by the mere replacement of r by n . 



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