206 THE REV. T. P. KIRKMAN ON NON-MODULAR GROUPS. 



the entire group of 120^ made with 12345, the sixty posi- 

 tive substitutions. Vide the preceding volume^ p. 283. 



Take now the ten elements 1234567890. First we are 

 to take aa^a^ mutually permutable, and such that bbi 

 being permutable, ab^ shall be of the fifth, and ab of the 

 third order. Mutually permutables are either 



1234567890 1 1234567890 ^ 



i325468709=:«^ I i325468709 = « I 



1543260987 = ^) ^^' 63245i0987 = «, | ^^ 

 1452369078 = («,), J 62354i9078 = «^;J 



and any of the substitutions aa^aj^a^{a^ may form part of 

 a didymous system of 5, as may be seen if we form the 

 didymous radicals of 2345178906. Any of them may also 

 form part of a didymous system of 3. This requires to be 

 shown thus. The following, 



1234567890 1324657980 

 2315648970 3216549870 

 3126459780 2135468790, 



is a group of 6 of my theorem G (art. 26). The didymous 

 radicals have here all four elements undisturbed. If now 

 we exchange the two last elementary groups thus, we have, 

 (art. 46) of my memoir in the last volume, 



132 798 465 o 

 321 987 654 o 

 213 879 546 o, 



a system of didymous radicals of the same substitution 

 2315648970, and all of the form oi aa^a^, which have only 

 two elements undisturbed. 



The above two groups H H' are curious, as proving that 

 two groups may be alike in the number and orders both of 

 their substitutions and of their circular factors, and yet not 

 be equivalent groups ; for every equivalent of the first will 

 have two vertical rows of elements undisturbed. 



