Functions formed on Groups. 23 



The 143 six-letter functions given by the first transitive 

 maximum groups of six letters, with full exhibition 

 of the values of the functions. By the Rev. Thos. P. 

 Kirkman, M.A., F.R.S. 



{Received October 20th, i8gi.) 



The groups to be treated are the elementary but highly 

 important ones of my §6, p. 301, Manchester Memoirs, 1862. 

 They are easily formed for every number n of letters, are 

 all of the order 2«, and all of the form G + RG, where G is 

 a transitive group of n powers 1,6, 2 ,0 S . ., whose derived 

 derangement RG is composed of n square roots of unity. 

 It happens in the cases of n—6 and «=4 that they are 

 also maximum groups, (F.G. 4). 



To form such a group of order 2;/, we complete under 1, 

 in unity =i2$...n t any vertical permutation C of the n 

 elements, and then complete the same circle C under every 

 element. This gives G. 



For RG, we write the same C again under 1 by itself, and 

 complete C vertically n-\ times more under the other 

 elements by this rule— In the horizontal line beginning 

 with e, plant your 1 at the e th place. 



For comparison of two given series of Q columns or 

 rectangles, each containing a function of n letters and its 

 Q— 1 values, i.e., for comparison of 



$+•• and <&••', 

 it is necessary that <$ a and O b should both exhibit unity 

 under exactly the same line 2 of n arbitrary Greek ex- 

 ponents, all different or not all. Then the two series are 

 one and the same algebraically, when and only when every 

 value of (3+- • is the same sum of L products of powers, as 

 is some value of Ot " » 



